Digital Communication

IV semester B. Tech. ECE

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Textbooks

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Introduction

• What is communication?
• Transfer of message from a source to a destination
• Classified as analog and digital communications
• Requirements: Signal, Transmission channel, Modulation, Coding
• Challenges: Bandwidth and Noise
• Analysis: SNR, Probability of error and Power spectra
• Design Concepts: Fourier Transform, Probability theory, Random process, Electronics and Digital signal processing and Information theory

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Introduction

• Message signals
• Classified as analog signals and digital signals
• Classified as periodic and aperiodic signals
• Examples of analog signals:
• speech (but words are discrete)
• music (closer to a continuous signal)
• temperature readings, barometric pressure, wind speed
• images stored on film
• AM and FM radio signals (generated by broadcast systems)
• Analog signals can be represented (approximately) as bits

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Introduction

• Message signals (contd.)
• Examples of digital signals:
• Ethernet, FAX
• Computer (microcontroller) generated data
• Chip level communication
• Examples of communication channels
• Radio (20 KHz to 20+ GHz):
• Coaxial cables, Twisted pair of copper wires, Wireless
• Optical fiber (200 THz or 1550 nm)

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Introduction

• Early works in Electronics
• John Abbrose Eleming (1904) invented the vacuum-tube diode
• Lee de Forest (1906) invented the vacuum-tube triode
• Amplifier for analog signals
• Walter H. Brattain, William Shockley of Bell Lab. (1948) invented the transistor
• Robert Noyce (1958) produced the first silicon integrated circuit (IC)

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Introduction

• Notable inventions
• Samuel Morse (1844) invented Telegraph
• Alexander Graham Bell (1875) invented the telephone
• A. B. Strowger (1897) devised the automatic step-by-step switch
• BBC (1939) started broadcasting television service on a commercial basis

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Introduction

• Wireless communication
• James Clerk Maxwell (1864) formulated the electromagnetic theory of light and predicted the existence of radio waves
• Heinrich Hertz (1887) demonstrated the existence of radio waves experimentally
• Oliver Lodge (1894) demonstrated wireless communication over a relatively short distance (150 yards)
• J. C. Bose (1895) demonstrated wireless communication over a distance of a mile using microwaves

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Introduction

• Wireless communication (contd.)
• Guglielmo Marconi (1901) demonstrated wireless communication over a long distance (2100 miles)
• Reginald Fessenden (1906) conducted the first radio broadcast using a technique called amplitude modulation
• Edwin H. Armstrong invented the superheterodyne radio receiver (1918) and demonstrated frequency nodulation (1933)

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Introduction

• Early works in Digital communication
• Harry Nyquist (1928) developed the theory of signal transmission in telegraphy
• Alex Reeves (1937) invented pulse-code modulation
• Claude Shannon (1948) laid the theoretical foundation of digital communications by proving that the probability of error increases only when the transmission rate exceeds the channel capacity.
• The theorems of Shannon inspired many signal compression algorithms and error correcting codes
• This new field has come be known as Information theory

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Introduction

• Early works in Digital communication (contd.)
• First call through a stored-program system was demonstrated in Bell Lab. (1958)
• The first commercial telephone service with digital switching in Morris, Illinois (1960)
• The first T-1 carrier system transmission was installed in Bell Lab. (1962)
• D. O. North (1943) developed matched filter for the optimum detection of a unknown signal in a additive white noise

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Introduction

• Internet
• From 1950s to 1970s various studies were made on computer networks
• In 1985, the Advanced Research Projects Agency Network (ARPANET) developed by US dept. of Defense became the Internet
• In 1990, Tim Bernes-Lee proposed a hypermedia software interface called “world wide web”
• By 1994, Internet became commercial entity

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Uses of Communication

• Cordless telephony
• Cellular communications
• Internet, Bluetooth, WiFi, etc.
• Satellite communication
• Industrial automation
• Vehicular applications (Navigation, object detection, infotainment and autonomous driving)
• E-commerce applications
• Biomedical, Sports, Animation, Augmented reality, Virtual reality
• Tracking services (human, livestock, cargo, ships, etc.)

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The Communication Process

• Communication is a step-by-step process of transmitting information from point-to-point
1. The generation of a message signal – voice, music, picture, or computer data.
2. The description of that message signal with a certain measure of precision, using a set of symbols – electrical, aural, or visual.
3. The encoding of those symbols in a suitable form for transmission over a physical medium of interest.
4. The transmission of the encoded symbols to the desired destination.
5. The decoding and reproduction of the original symbols.

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The Communication Process

• The re-creation of the original message signal with some definable degradation in quality, the degradation being caused by unavoidable imperfections in the system.
2. Basic elements of communication

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Digital Communication

• Key functional blocks include the following
• Source encoder–decoder
• Channel encoder–decoder
• Modulator–demodulator
• The source encoder removes redundancy from the message signal for efficient use of the channel
• The resulting sequence of symbols is called source codeword

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Digital Communication

• The channel encoder introduces controlled redundancy into the data stream to improve the reliability of the transmission
• The resulting sequence of symbols is called channel codeword
• The modulator represents each symbol of the channel codeword by a corresponding analog symbol
• The resulting sequence of analog symbols is called waveform
• At the receiver, the channel output (received signal) is processed in reverse order to that in the transmitter

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Information Theory – Introduction

• In 1948, the theoretical foundations of DC were laid down by Claude Shannon in a paper entitled “The mathematical theory of communication”
• His work proved that the probability of error increases only when the transmission rate exceeds the channel capacity
• His work gave rise to Information theory whose foundations lie in probability theory
• This ground-breaking result also gave rise to error correcting codes aka channel coding (Golay in 1949, Hamming in 1950, LDPC codes Turbo codes in 1993)

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Information Theory – Introduction

• Information theory deals with mathematical modeling and analysis of a communication system rather than with physical sources and physical channel
• Two fundamentals questions addressed are
• What is the irreducible complexity, below which a signal cannot be compressed?
• What is the ultimate transmission rate for reliable communication over a noisy channel?
• The answers lie in Entropy (former) and Capacity (latter)

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Information Theory – Introduction

• Let us focus on the information rather than a signal
• What is information?
• A random signal carries information (i.e., surprise or uncertainty) compared to a deterministic signal.
• Consider a source which randomly emits one symbol in each signaling interval from the alphabet 𝒮={s₀,s₁,…s_K-1}
• Let the probability of symbol s_k being emitted be given by P(S=s_k) = p_k
• The amount of information gained by resolving the uncertainty (i.e., observing the symbol) is inversely proportional the probability of occurrence of the symbol

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Information Theory – Introduction

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Information Theory – Introduction

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Information Theory – Introduction

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Information Theory – Introduction

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Information Theory – Introduction

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Information Theory – Introduction

• Example: Entropy of Bernoulli random variable

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Information Theory – Introduction

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Information Theory – Introduction

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Information Theory – Introduction

• Example of extended source (contd.)

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Information Theory – Source coding

• The process of representing the data generated by a discrete source of information is called source encoding
• The device which provides such representation is called source encoder
• It is desirable to know the statistics of the source.
• Source encoder assigns short codewords to frequent source symbols and long codewords to rare source symbols
• This type of source code is called variable-length code

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Information Theory – Source coding

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Information Theory – Source coding

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Information Theory – Source coding

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Information Theory – Lossless data compression

• Signals produced by physical sources have redundancy which can be removed prior to transmission
• Lossless data compression deals with removal of redundancy in the transmitted signal
• It reduces the number of bits per source symbol
• It allows successful reconstruction of the original data
• Entropy establishes the fundamental limit on lossless data compression
• Examples of lossless data compression:
• Prefix coding, Huffman coding, Lempel-Ziv coding

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Information Theory – Prefix coding

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Information Theory – Prefix coding

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Information Theory – Prefix coding

• Illustration:
• The source decoder starts with the beginning of the sequence and decodes one codeword at a time
• A decision tree is navigated from the initial state to one of the terminal states
• When a terminal state is reached, the corresponding symbols is emitted

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Information Theory – Prefix coding

• Given that code II is the prefix code, use the decision tree below to decode the received binary sequence 1011111000

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Information Theory – Prefix coding

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Information Theory – Prefix coding

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Information Theory – Prefix coding

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Information Theory – Huffman coding

• Huffman coding algorithm computes the optimal prefix code for a given distribution such that the average codeword length is close to the entropy of the discrete memoryless source.
• Algorithm
• The source symbols are listed in order of decreasing probability.
• Splitting stage: The two source symbols of lowest probability are assigned 0 and 1.
• These two source symbols are then combined into a new source symbol with probability equal to the sum of the two original probabilities.

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Information Theory – Huffman coding

• Algorithm (contd.)
• The probability of the new symbol is placed in the list in accordance with its value.
• The procedure is repeated until we are left with a final list of source statistics (symbols) of only two for which the symbols 0 and 1 are assigned
• The code for each (original) source is found by working backward and tracing the sequence of 0s and 1s assigned to that symbol as well as its successors.
• The algorithm does not produce unique code because of the arbitrariness in the way the symbols 0 and 1 are assigned when probabilities are equal

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Information Theory – Huffman coding

• Example: Consider the five symbols of the alphabet of a discrete memoryless source and their probabilities given below.

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• Construct a Huffman tree and find the Huffman code. Verify Kraft inequality.
• Solution

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Information Theory – Huffman coding

• Solution (contd.)

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• Place the sum probability above the symbol probability

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Information Theory – Huffman coding

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Information Theory – Lempel-Ziv coding

• Huffman coding requires prior knowledge of the source symbol probabilities
• In the modeling of text, the storage requirements for the Huffman code grow exponentially as the symbols generated by the extended source increase
• The above drawbacks are overcome using the Lempel-Ziv coding which is now a standard technique
• The source data stream is parsed into segments that are the shortest sub-sequences not encountered previously
• Lempel-Ziv codes are fixed-length codes. The last bit is called the innovation symbol

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Information Theory – Lempel-Ziv coding

• Illustration: Consider encoding the binary data sequence 000101110010100101
• Step 1: Assume the subsequences 0 and 1 are already stored in the codebook
• Step 2: Scan for the next shortest subsequence from the left. Store it, remove it and and repeat step 2.

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• New subsequence is 00 which is stored into the codebook

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Information Theory – Lempel-Ziv coding

• New subsequence is 01 which is stored into the codebook

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• New subsequence is 011 which is stored into the codebook

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• In this manner, identify all unique sub-sequences till the end of the data is reached
• Step 3: For each subsequence (except 0 and 1), identify the root subsequence that matches and create the numerical representation.

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Information Theory – Lempel-Ziv coding

• Step 3 (contd.): The last digit in the numerical representation corresponds to the numerical position of innovation bit (last bit in a subsequence)

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• Example: Numerical representation of subsequence 00 in position 3 is 11 which is obtained as follows.
• Subsequence 00 contains the subsequence 0, hence the leftmost digit is 1. The innovation bit is 0, hence the rightmost digit is 1

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Information Theory – Lempel-Ziv coding

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• Example: Numerical representation of subsequence 100 in position 8 is 61 which is obtained as follows.
• Subsequence 100 contains the subsequence 10, hence the leftmost digit is 6. The innovation bit is 0, hence the rightmost digit is 1
• Example: Numerical representation of subsequence 101 in position 9 is 62 which is obtained as follows.
• Subsequence 101 contains the subsequence 10, hence the leftmost digit is 6. The innovation bit is 1, hence the rightmost digit is 2

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Information Theory – Lempel-Ziv coding

• Step 4: Converting the numerical representation into binary encoded blocks

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• For the given numerical representation, a three digit binary form is sufficient to represent the remaining digits to the left. The innovation bit gives the rightmost bit.
• Concatenate the above to form the codeword.
• In practice, 12-bit long fixed blocks are used. Therefore, the codebook has 2¹² = 4096 entries

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Information Theory – Lempel-Ziv coding

• Step 4 (contd.):

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• Example: Numerical representation 21 is converted into 0100 as follows.
• Innovation bit corresponding to 1 is 0 and binary form of remaining digits (i.e., 2) is 010. Hence, we get 0100
• Example: Numerical representation 62 is converted into 1101 as follows.
• Innovation bit corresponding to 2 is 1 and binary form of remaining digits (i.e., 6) is 110. Hence, we get 1101

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Information Theory – Discrete memoryless channels

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Information Theory – Discrete memoryless channels

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Information Theory – Discrete memoryless channels

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Information Theory – Discrete memoryless channels

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Information Theory – Mutual information

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Information Theory – Mutual information

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Information Theory – Mutual information

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Information Theory – Mutual information

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Information Theory – Mutual information

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Information Theory – Mutual information

• Proof:

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Information Theory – Mutual information

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Information Theory – Mutual information

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Information Theory – Mutual information

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Information Theory – Mutual information

• Properties of mutual information (contd.):
• Relation between various channel entropies

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Information Theory – Channel capacity

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Information Theory – Channel capacity

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Information Theory – Channel capacity

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Information Theory – Channel capacity

• Example: Capacity of binary symmetric channel

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Information Theory – Channel capacity

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• Channel capacity attains maximum value of 1 bit when channel is noise free
• Channel capacity attains minimum value of 0 when probability of error is 1/2

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Information Theory – Channel coding theorem

• Channel encoder maps the input data sequence to the channel input sequence
• Channel decoder maps the channel output sequence to the output data sequence
• The above components ensure that the impact of the channel noise on the digital communication system is minimized (i.e., reliability is maximized)

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Information Theory – Channel coding theorem

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Information Theory – Channel coding theorem

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Information Theory – Channel coding theorem

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Information Theory – Channel coding theorem

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Information Theory – Differential entropy

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Information Theory – Differential entropy

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Information Theory – Differential entropy

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Information Theory – Relative entropy for continuous random variables

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Information Theory – Relative entropy for continuous random variables

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Information Theory – Relative entropy for continuous random variables

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Information Theory – Mutual information of continuous r.v.

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Information Theory – Mutual information of continuous r.v.

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Information Theory – Information capacity law

• Consider a discrete time memoryless, band-limited, power-limited Gaussian channel
• X(t) is sampled every 1/2B seconds
• Samples are transmitted over noisy channel every T seconds (i.e., total samples in T, denoted by K = 2BT)

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Information Theory – Information capacity law

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Information Theory – Information capacity law

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Information Theory – Information capacity law

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Information Theory – Information capacity law

• Information capacity law can be stated as

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• It is easier to increase the information capacity of a continuous communication channel by expanding its bandwidth than by increasing the transmit power for a prescribed noise variance
• Given a fixed transmit power P and channel bandwidth B, the information capacity of C bits-per-second can be achieved with arbitrarily small probability of error by choosing a sufficiently complex encoding system

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Information Theory – Information capacity law

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Information Theory – Information capacity law

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Information Theory – Rate distortion theory

• For source coding to be perfect (i.e., lossless compression), the average codeword length must be at least equal to the entropy of the source alphabet.
• However, this could be constrained
• The communication channel may pose limits on the permissible code rate and therefore, on the average codeword length
• In representing continuous signals (e.g., speech as PCM), information transmission rate may exceed channel capacity
• In such cases, the source coding will be lossy
• Shannon’s coding theorem in such cases leads to the study of rate distortion theory

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Information Theory – Rate distortion theory

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Information Theory – Rate distortion theory

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Information Theory – Rate distortion theory

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Information Theory – Rate distortion theory

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Review of Fourier Analysis

• Implications of Fourier Analysis
• The Fourier transform of a signal specifies the complex amplitudes of the components that constitute the frequency-domain description or spectral content of the signal.
• The inverse Fourier transform uniquely recovers the signal, given its frequency-domain description.
• The Fourier transform is endowed with several important properties, which, individually and collectively, provide invaluable insight into the relationship between a signal defined in the time domain and its frequency domain description.
• A signal can only be strictly limited in the time domain or the frequency domain, but not both.
• Bandwidth is an important parameter in describing the spectral content of a signal and the frequency response of a linear time-invariant filter

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Review of Fourier Analysis

• The Fourier transform and its inverse were represented as

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• This representation based on jω was used to make all transforms (Fourier, Laplace, DTFT, z) similar in form
• Hereafter, we will use 2πf instead of ω as it simplifies a lot of transforms and theorems

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Review of Fourier Analysis

• Therefore, we have the Fourier transform and its inverse given as

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Review of Fourier Analysis

• Properties of Fourier transform
• Duality

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• Linearity (superposition)

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• Dilation or time scaling

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• Conjugation

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Review of Fourier Analysis

• Example 1: Apply duality and dilation to prove the Fourier transform pair given below

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• Hint: Assume a=2W is the bandwidth

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Review of Fourier Analysis

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Example 1 (contd.)

Review of Fourier Analysis

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Review of Fourier Analysis

• Example 1 (contd.): Time and frequency domain representations of the sinc pulse

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Review of Fourier Analysis

• Properties of Fourier transform (contd.)
• Time shifting

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• Frequency shifting

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• Differentiation in time domain

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• Integration in time domain

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where

Review of Fourier Analysis

• Properties of Fourier transform (contd.)
• Modulation theorem

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• Convolution theorem

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• Parseval’s theorem or Rayleigh energy theorem

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Review of Fourier Analysis

• Homework: Find Fourier transform of rectangular pulse

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• Hint: Apply time shift of T/2 and use the relation

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• Your result should be

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Review of Fourier Analysis

• Homework: Find Fourier transform of a radio frequency (RF) pulse

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• Either solve directly or use the relation and apply frequency shift property

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• Your result should be

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Review of Fourier Analysis

• Radio Frequency (RF) Pulse (contd.)

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Analog communication

• Why analog communication?
• Few analog communication systems are still in use
• E.g., Broadcast systems
• Mixers and passband filters are still in use
• Real-time operations
• At very high frequencies, ADC/DAC operation becomes difficult
• e.g., 5G systems where hybrid analog/digital techniques are used

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Analog communication

• Baseband signal
• The frequency band of the information bearing signal is called the baseband
• Examples of baseband transmission
• Telephones: 300-3700 Hz
• High-fidelity audio: 0-20 KHz
• Television (NTSC) video: 0-4.3 MHz
• Ethernet (10 Mbps): 0-20 MHz
• Baseband transmission usually requires wire

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Analog communication

• Passband transmission
• This transmission is achieved by modulation
• Modulation is a process by which a sinusoidal high frequency signal (carrier) is varied with respect to the amplitude of the baseband signal (modulating signal)
• Wireless communication requires frequencies higher than baseband
• Multiple signals can be sent at same time using different frequencies:
• This is known as frequency division multiplexing (FDM)
• Alternately, multiple baseband signals can share the same spectra using time division multiplexing (TDM)

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Analog communication

• Passband transmission (contd.)
• Let the baseband signal be denoted by m(t).
• The signal m(t) may modify (i.e., modulates) either the amplitude, frequency, or phase of carrier signal.
• Amplitude modulation: A(t) is proportional to m(t)
• Frequency modulation: f_c(t) is proportional to m(t)
• Phase modulation: φ(t) is proportional to m(t)
• Frequency and phase modulation are also called more generally as angle modulation
• The resulting passband signal s(t) is expressed as

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Analog communication

• Need for modulation
• Efficiency of transmission increases with frequency
• For efficient line-of-sight transmission of electromagnetic waves, antenna dimensions must be at least 1/10 of signal’s wavelength
• To overcome hardware limitations
• Hardware cost is minimized when fractional bandwidth (i.e., absolute bandwidth divided by center frequency) should be 1-10%
• To reduce the noise and interference (instead of increasing power, increase bandwidth)
• Simultaneous baseband signals can be separated by assigning separate carrier frequencies

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Analog communication

• Example 2: Find the required antenna height if f₀=2 kHz. Assume antenna height is λ/4
• λ =c/f₀ = 150 km. Hence. antenna height will be 37.5 km. Impractical!
• Example 3: Consider the speech signal that occupies the band from 300 Hz to 3 kHz. Comment on the antenna height
• When f₀ = 300 Hz, antenna height is 250 km. When f₀ = 3 kHz, antenna height is 25 km.
• Single antenna won’t be able to handle the signal!

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Amplitude modulation (AM)

• In AM, the continuity of the carrier is maintained in time, hence such schemes are also known as continuous wave (CW) or linear modulation
• The amplitude of the carrier is varied according to the amplitude of the message signal
• Let carrier be denoted by
• The AM wave is described as s(t) given below

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• The parameter k_a is called amplitude sensitivity
• Here, µ=|k_a m(t)|_max is called the modulation index

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Amplitude modulation (AM)

• Effect of modulation index

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Modulation index must be kept less than 1

Amplitude modulation (AM)

• In AM, information pertaining to the message signal m(t) resides solely in the envelope of s(t)
• Envelope is defined as the amplitude of the modulated wave. It is given by Ac|1+k_am(t)|
• The envelope has a similar shape as m(t)
• µ should be less than 1 to recover the m(t)
• When µ > 1, the carrier is said to be over-modulated, resulting in phase reversals
• Carrier frequency should be much greater than W

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Amplitude modulation (AM)

• Frequency domain representation
• Given the modulated signal is

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• The Fourier transform gives

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• Notice, that the spectral components corresponding to ±f_c are also present besides the side bands of width 2W
• The spectral components corresponding to the message signal also appear, albeit, attenuated compared to m(t)

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Amplitude modulation (AM)

• Frequency domain representation

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Amplitude modulation (AM)

• Example 7: Single-tone modulation
• Consider a modulating signal m(t).
• The AM wave is given by

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• The Fourier transform of AM wave

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Amplitude modulation (AM)

• Example 7: Single-tone modulation (contd.)

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Amplitude modulation (AM)

• Example 7: Single-tone modulation (contd.)
• Let A_max and A_min denote the maximum and minimum values of the envelope in s(t)
• The modulation index can be expressed as

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• Power of AM wave across 1 Ohm resistance
• Carrier power + Upper side-frequency power + Lower side-frequency power

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Amplitude modulation (AM)

• Transmission efficiency (i.e., ratio of the total sideband power to the total power) for the single tone modulated wave is given by

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For µ=1, the transmission efficiency η is 33%

Amplitude modulation (AM)

• Transmission efficiency (η): General case

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Amplitude modulation (AM)

• Transmission efficiency (η): General case (contd.)

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Amplitude modulation (AM)

• Square law modulator: To generate AM
• A square-law modulator for generating an AM wave relies on the use of a nonlinear device (e.g., diode)

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Amplitude modulation (AM)

• Square law modulator (contd.)
• Ignoring the higher order terms, the input-output characteristic of the diode-load resistor combination is represented by the square law:

where , a₁ and a₂ are

constants

• Substituting v₁(t) in v₂(t) gives

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Amplitude modulation (AM)

• Square law modulator (contd.)
• The spectral components of v₂(t) are given by
• AM part assuming A_c=1:

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• Undesirable part:

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• A band-pass filter with cut-off frequencies f_c-f_m and f_c+f_m can remove the undesirable part

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Amplitude modulation (AM)

• Envelope detector
• One of the simplest and economic circuits
• Note the AM modulator itself is a very simple circuit
• The detector is feasible provided the following conditions are satisfied
• The AM wave is narrowband (i.e., carrier frequency is large compared to the message bandwidth)
• The percentage modulation (i.e., µ×100%) in the AM wave is less than 100 percent
• The circuit used here for envelope detector is a series type consisting of diode and RC filter

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Amplitude modulation (AM)

• Envelope detector (contd.)

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Amplitude modulation (AM)

• Envelope detector operation
• Assume the diode is ideal
• That is, it offers r_f resistance in forward bias and infinite resistance in reverse bias
• For positive half cycle of the input signal, the diode is forward biased and the capacitor C charges to the peak voltage
• For negative half cycle, the diode is reverse biased and the capacitor C discharges
• The rate of discharge is controlled such that it is lesser than f_c
• This cycle of operation repeats with diode acting as a switch performing ON-OFF cycles

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Amplitude modulation (AM)

• Rate of charge:

Resistance of RC circuit during forward bias is R_s+r_f where R_s is internal resistance of voltage source

So the charging time is (R_s+r_f)C

• Rate of discharge:

Resistance of RC circuit during reverse bias is the load resistance R_l

So the discharging time is R_lC

• Condition for charging Condition for discharging

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Amplitude modulation (AM)

• Example for envelope detector

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AM problems

• From the given S(f), find s(t), μ, η, P_s

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AM problems

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AM problems

• If P_C is 90% of P_T find μ

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• Let s(t) = 5cos(450πt)+20cos(550πt)+5cos(650πt)
• Find the following: f_c, f_m, A_c, μ, P_T and η. Also draw the magnitude spectrum S(f) and power spectrum

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AM problems

s(t) = 5cos(450πt)+20cos(550πt)+5cos(650πt)

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AM problems

• For the given power spectrum find μ, P_T and η

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AM problems

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AM problems

• Consider the AM signal where α > 0

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Show that to avoid envelope distortion α ≤ 8/9

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To avoid envelope distortion, we need the envelope A(t) ≥0

In other words

AM problems

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DSB-SC

• Double sideband modulation – Suppressed carrier (DSB-SC)
• This is the simplest modulation technique
• Based on frequency shift property of Fourier transform
• Unlike AM, the carrier is suppressed resulting in better transmission power efficiency
• Let m(t) denote the message signal
• Let c(t) = A_ccos(2πf_ct) denote the sinusoidal carrier signal
• Let s(t) denote the modulator output
• The modulator is referred to as product modulator

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DSB-SC Modulation

• DSB-SC Modulator:
• Time and frequency domain representations

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• The Fourier transform of the modulator output is given by

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DSB-SC Modulation

• Derivation of the Fourier transform of the DSB-SC signal

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DSB-SC Modulation

• Time and frequency domain representations

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DSB-SC Modulation

• DSB-SC Modulator (contd.)
• The Fourier transform of the modulator output is given by

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• Example 4: Suppose the message signal is a sinusoidal signal of amplitude A_m and frequency f_m, then the Fourier transform of the message signal is given by

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• Then, the modulator output S(f) can be decomposed into 4 spectral components

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For A_c=1 and A_m=1, see the frequency domain representation of the DSB-SC wave

in the next slide

DSB-SC Modulation

• DSB-SC- Modulator (contd.)

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• Note:
• Deterministic signals do not contain information.
• Only random signals do. Hence, we need to consider random process and their power spectra.
• The above deterministic example is for illustration purpose only.

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DSB-SC Modulation

• Example 5: For the single tone DSB-SC modulation considered in previous slide
• What is the average power of the DSB-SC modulated wave?
• Solution

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DSB-SC Modulation

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• Solution (contd.)

DSB-SC Modulation

• Solution (contd.)
• Average power of the DSB-SC wave based on s(t): Average power of carrier × Average power of message

​

• Average power of the DSB-SC wave based on S(f): Average power per delta component × No. of delta components

​

​

​

​

​

155

DSB-SC Modulation

• Example 5 (contd.): Suppose the sinusoidal carrier c(t)=A_ccos(2πf_ct) is modulated by message signal m(t)=A_msin(2πf_mt) then express the Fourier transform of the DSB-SC modulated wave s(t).
• What is the average power of the DSB-SC modulated wave?

156

DSB-SC Modulation

• Solution:

157

DSB-SC Modulation

• Solution (contd.):

158

DSB-SC Modulation

• DSB-SC: Carrier versus bandwidth
• Transmitters can radiate only a narrow band without distortion. Therefore, the carrier frequency is chosen such that

159

DSB-SC Modulation

• Generation of DSB-SC
• Multiplier modulators using variable gain amplifiers
• Nonlinear modulator: Suppose the input-output characteristic is

​

• Let

​

​

• Suppose we apply x₁(t) and x₂(t) to the non-linear modulator, and look at the difference

160

DSB-SC Modulation

• Generation of DSB-SC
• Nonlinear modulator (contd.):

​

​

​

​

​

• The unwanted baseband component is blocked by bandpass filter. This could be the antenna or the amplifier.

161

DSB-SC Modulation

• Generation of DSB-SC
• Nonlinear modulator (contd.): Schematic

​

​

​

​

​

• Homework: Consider a device whose output is the square of the input (e.g., diode, transistor, etc.). Suppose the input is sum of c(t) = A_ccos(2πf_ct) and m(t). Draw the schematic for the modulator and draw the frequency domain representation of the output.

162

DSB-SC Modulation

• Average power of the DSB-SC waveform
• Let m(t) have an average power of P_m Watts

​

163

DSB-SC Demodulation

• Demodulation of DSB-SC: Coherent detection
• Schematic and frequency response of v(t)

164

DSB-SC Demodulation

• Demodulation of DSB-SC: Coherent detection
• Both modulator and demodulator use a multiplier having carrier as an input
• Modulator uses bandpass filter
• Demodulator uses lowpass filter
• The carrier used by the demodulator must be in phase with the transmitter carrier
• Such a receiver is called synchronous, coherent, homodyne.
• The receiver has a local oscillator (VCO) that must be adjusted to stay in phase with the received signal
• Let the phase shift (in practice) in the carrier generated by the local oscillator be denoted by φ

165

DSB-SC Demodulation

• Demodulation of DSB-SC: Coherent detection
• The carrier from the VCO is represented as

​

• Upon multiplying the carrier with the received signal, we get

166

DSB-SC Demodulation

• Demodulation of DSB-SC: Coherent detection
• The first term of v(t) represents a new DSB-SC modulated term with carrier frequency of 2f_c
• The second term of v(t) is proportional to the baseband signal m(t)
• The first term can be removed by a low pass filter
• Cut-off frequency of LPF must be greater W but less than 2f_c –W
• Thus the demodulated output is given by

167

DSB-SC Demodulation

• Demodulation of DSB-SC: Coherent detection
• Observations:
• When phase error φ is constant, then v₀(t) is proportional to m(t)
• Amplitude of demodulated signal is maximum when φ=0 and zero when φ = ± π/2.
• The zero demodulated signal, which occurs for φ = ± π/2 represents the quadrature null effect, which is an inherent property of coherent detection
• The phase error in the local oscillator causes the detector output to be attenuated by a factor equal to cos(φ)
• The phase error fluctuates randomly in practice!

168

DSB-SC Demodulation

• Example 6: The sinusoidally modulated DSB-SC wave (from Example 4) is applied to a product modulator using a locally generated sinusoid of unit amplitude, and which is synchronous with the carrier used in the modulator.
• Determine the output of the product modulator, denoted by v(t)
• Identify the spectral components in v(t) and draw the frequency domain representation.
• What will be the LPF cut-off frequency?

169

DSB-SC Demodulation

• Example 6 (soln.):
• The DSB-SC modulated wave (from Example 4) which is input to the coherent detector is

​

​

​

​

​

• After multiplying AM wave with the local oscillator output of the coherent detector, we get

170

DSB-SC Demodulation

• Example 6 (soln.):

171

DSB-SC Demodulation

• Example 6 (soln.):

172

DSB-SC Demodulation

• Example 6 (soln.):

​

​

​

​

173

Angle modulation

• The angle of the carrier wave is varied according to the amplitude of the message signal
• Angle modulation is a non-linear process
• In analytic terms, the spectral analysis of angle modulation is complicated.
• In practical terms, the implementation of angle modulation is demanding
• Angle modulation does not have bandwidth restrictions as in amplitude modulation
• As carrier amplitude will be constant in angle modulation, effect of additive noise is limited

174

Angle modulation

• Let θ_i(t) denote the angle of a modulated sinusoidal carrier at time t
• θ_i(t) is a function of the message signal m(t)
• An angle modulated wave is represented as

​

• A_c is the carrier amplitude
• A complete oscillation occurs whenever θ_i(t) changes by 2π radians

​

​

175

Angle modulation

• Suppose θ_i(t) increases monotonically with time, the average frequency in Hertz, over a small interval of t to t+∆t is given by

​

​

• Allowing ∆t tend to zero gives the definition of the instantaneous frequency

176

Angle modulation

• Thus, the modulated signal s(t) can be interpreted as a rotating phasor of length A_c and angle θ_i(t)
• The angular velocity of the phasor is
• Measured in radians/sec
• For an unmodulated carrier signal the instantaneous angle θ_i(t) is given by

​

• There are infinite number of ways to change the angle according to the message signal

177

Angle modulation: Basic definitions

• Phase modulation (PM) is that form of angle modulation in which the instantaneous angle θ_i(t) is varied linearly with the message signal m(t)

​

​

• The first term represents the angle of the unmodulated carrier
• The second term represents the variable part due to the modulating signal
• The term k_p is called the phase sensitivity factor, measured in radians/volt

178

Angle modulation: Basic definitions

• Phase modulation (PM) (contd.)
• The PM wave is given by

​

• Frequency modulation (FM) is that form of angle modulation in which the instantaneous frequency f_i(t) is varied linearly with the message signal m(t)

​

• Here, f_c is the frequency of the unmodulated carrier
• k_f is the frequency sensitivity factor measured in Hertz/volt

179

Angle modulation: Basic definitions

• Frequency modulation (FM) (contd.)
• Integrating the instantaneous frequency with respect to time and multiplying by 2π we get

​

​

​

​

• The FM wave is given by

180

Angle modulation: Basic definitions

181

Relation between PM and FM

182

Relation between PM and FM

• Example: PM using FM modulation
• Let define a modulating signal
• The instantaneous angle for FM is

​

• But replacing the input m(t) by gives

​

• The corresponding FM modulator output is given by

​

• However, the above equation resembles a PM wave whose phase sensitivity is denoted by k_f (instead of k_p)

183

Relation between PM and FM

• Example: FM using PM modulation
• Let define a modulating signal
• The instantaneous angle for PM is

​

• The corresponding PM modulator output is given by

​

​

• However, the above equation resembles a FM wave whose frequency sensitivity is denoted by k_p (instead of k_f)

184

Angle modulation

185

Carrier wave

Message signal

AM wave

Angle modulation

186

Carrier wave

Message signal

PM wave

Angle modulation

187

Carrier wave

Message signal

FM wave

Angle modulation

188

Angle modulation

• Example: Consider the message signal m(t) given below. Let f_m=1 kHz, f_c =100 MHz, the values of k_p for FM and PM are 2π·10⁵ and 10π respectively

​

189

Angle modulation

• Instantaneous frequency in FM

​

​

​

​

• The instantaneous frequency changes linearly between the minimum and maximum frequencies for each half cycle

190

Angle modulation

• Instantaneous frequency in PM

​

​

​

​

• Because ṁ(t) switches back and forth from min to max, the carrier frequency switches back and forth between minimum and maximum frequencies each half cycle

191

Properties of angle modulated waves

• Constancy of transmitted power
• Average power transmitted by PM/FM wave is A_c²/2
• Nonlinearity of modulation process
• Consider a message signal m(t)=m₁(t)+m₂(t)
• The PM wave is given by

​

• Suppose we modulate separately

​

​

• It can be seen that superposition principle doesn’t hold

192

Properties of angle modulated waves

• Irregularity of zero-crossings
• The information content of an angle modulated wave resides in its zero crossings
• As a consequence of making θ_i(t) depending on m(t) (as in PM) or integral of m(t) over a small interval τ (as in FM), the zero crossings in PM and FM have irregularity in the spacing across the time scale
• Two cases where regularity can be achieved
• When the message signal m(t) changes linearly with time t, then f_i(t) of the PM wave varies according to the slope of m(t)
• When message signal m(t) maintains a constant value +ve or –ve, then f_i(t) of the FM wave has a constant value of m(t)

193

Properties of angle modulated waves

• Visualization of message signals in angle modulated is difficult
• Unlike amplitude modulation, the envelope does not capture the message waveform
• Tradeoff of increased transmission bandwidth for improved noise performance
• The improvement in noise performance is at the expense of increased transmission bandwidth which cannot be afforded in AM

194

Frequency Modulation

• FM spectral analysis is complicated, hence we consider two simple cases
• First we study single tone modulation that produces a narrow band FM wave
• Next we study single tone modulation that produces a wide band FM wave
• We don’t go into multi-tone modulation as it is very difficult to interpret

195

Narrow band FM

• Consider a sinusoidal modulating signal defined by

​

• The instantaneous frequency of the resulting FM wave is

​

where

• The term ∆f is called frequency deviation
• It represents the maximum departure of the instantaneous frequency from the carrier frequency f_c
• ∆f is proportional to amplitude of m(t) and independent of f_m

196

Narrow band FM

• As the instantaneous angle and instantaneous frequency are related by

​

​

we get

​

where the ratio of ∆f to f_m is called the modulation index (denoted by β and measured in radians) of FM wave

197

Narrow band FM

• Hence, the instantaneous angle can be written as

​

• It can be observed that the term β represents the phase deviation of the FM wave
• In other words, it represents maximum departure of the instantaneous angle θ_i(t) from the angle 2πf_ct of the unmodulated carrier
• For FM to be narrow band FM, β must me less than 1
• The FM wave is now given by

​

198

Narrow band FM

• Use the trigonometric identity (given below) to find spectral components

​

​

​

​

• Under the condition that β is less than one radian, we use two approximations,

and

​

​

199

Narrow band FM

• Therefore, we have the narrow band FM wave given by

​

​

​

​

​

• Note that the approximated narrowband FM has amplitude distortion unlike the ideal FM
• Draw the spectrum for the above equation

200

Narrow band FM

• One sided spectrum is shown (other side is similar)

201

Narrow band FM

• Example: Suppose A_m= 1 V, f_m=1 kHz, f_c =100 MHz and k_f = 10⁵ Hz/V for the FM signal. Find the frequency deviation and phase deviation of the FM signal. Is the FM signal an NBFM?

​

• Example: Suppose the modulation index of an FM signal is 0.6, the peak amplitudes of m(t) and c(t) are 1V and the highest frequency of m(t) occurs at 1 kHz. Find the total power, frequency deviation and the frequency sensitivity factor.

202

Narrow band FM

• Alternative derivation

203

Narrow band FM

• Alternative derivation

204

Narrow band FM

• Alternative derivation

205

Narrow band FM

• Comparison of narrow band FM with AM
• Note that the narrow band FM wave is similar to the AM wave except for sign change and modulation index definition

206

NBFM and AM Phasor comparison

• AM Phasor

​

​

​

• Carrier phasor is used as reference
• Phasor for carrier has length A_c and rotates (counter-clockwise) with an angular velocity of 2πf_c rad/sec
• USB phasor has length µA_c/2 rotates with an angular velocity of 2 π(f_c+f_m) rad/sec (i.e., leads the carrier)
• LSB phasor has length µA_c/2 rotates with an angular velocity of 2 π(f_c-f_m) rad/sec (i.e., lags behind carrier)

207

NBFM and AM Phasor comparison

208

NBFM and AM Phasor comparison

• NBFM Phasor

​

​

​

• Carrier phasor is used as reference
• Resultant phasor has approximately same amplitude as the carrier phasor but is at right angle to the carrier phasor
• In other words, the carrier and FM signal are out of phase with each other
• This is the fundamental difference between AM and NBFM

209

NBFM and AM Phasor comparison

210

Narrow band FM

• Example: NBFM
• What is the ratio of the maximum to the minimum value of the envelope in NBFM?
• Determine the average power of the narrow-band FM wave, expressed as a percentage of the average power of the unmodulated carrier wave.
• By expanding the angular argument θ(t)=2πf_ct+φ(t) of the narrow-band FM wave s(t) in the form of a power series, and restricting the modulation index β to a maximum value of 0.3 radian, show that

211

Narrow band FM

• Example: NBFM (contd.)
• What is the value of the harmonic distortion for β =0.3 radian?
• Solution
• 1^st part:

​

• 2^nd part:

​

• 3^rd part: Using the power series expansion

212

Narrow band FM

• Solution (contd.)

​

213

NBFM generation

• The schematic is obtained from the NBFM wave equation corresponding to

214

Single tone FM for arbitrary β

• Spectral analysis of single tone FM

215

Single tone FM for arbitrary β

216

Single tone FM for arbitrary β

217

Single tone FM for arbitrary β

218

(8)

(9)

Single tone FM for arbitrary β

219

Single tone FM for arbitrary β

• For positive or negative (integer) values of n

​

​

• For small values of β (much less than 1 radian)

​

​

• For any arbitrary β, the following equality holds

220

Substituting these values into (8) or (9) in slide 218 gives NBFM

Single tone FM for arbitrary β

• Based on S(f) given in (9) on slide 218 and the properties of J_n(β), we have the following statements for single tone FM wave
• For arbitrary values of β, the spectrum contains infinite components located symmetrically on either side of the f_c at frequency separations of f_m, 2f_m, 3f_m,...
• Carrier amplitude depends on β (i.e., A_cJ₀(β))
• Average power is constant

221

Single tone FM for arbitrary β

• Effect of modulation index β

​

​

​

​

• In this case ∆f is varied keeping f_m constant
• Only positive frequencies are shown for illustration
• As β increases, ∆f increases adding more spectral lines with constant frequency spacing

222

Single tone FM for arbitrary β

• Effect of modulation index β

​

​

​

​

223

In this case fm is varied and ∆f is kept constant

Only positive frequencies are shown for illustration

As β increases, f_m decreases with more spectral lines getting accommodated within 2∆f (i.e., f_c – ∆f < f < f_c+ ∆f)

Transmission bandwidth of single tone FM waves

• Thus, for large β, the bandwidth approaches 2∆f
• And, for small β, the bandwidth approaches 2f_m
• Carson’s rule: The transmission bandwidth of FM wave modulated by a single tone modulating wave of frequency f_m is given by

​

​

• Alternatively, we can define 99% bandwidth

​

​

224

Numerical problem

225

Numerical problem

226

Numerical problem

227

Transmission bandwidth of single tone FM waves

• The number of significant side frequencies 2n_max varies with β
• The universal curve for evaluating the normalized bandwidth is used for practical purposes
• For non-sinusoidal modulating wave, we take the ratio of ∆f to the highest frequency B
• This ratio is called deviation ratio (denoted by D= ∆f/B)
• Then the generalized Carson rule is B = 2(∆f +B)
• Next two slides gives the table for n_max and β and also universal figure for 99% bandwidth

228

Transmission bandwidth of single tone FM waves

229

Transmission bandwidth of single tone FM waves

230

Transmission bandwidth of single tone FM waves

231

Transmission bandwidth of single tone FM waves

• For the m(t) shown below, the spectral content till the 5^th harmonic is assumed to be significant. If k_f = 100 kHz/volt, what is the bandwidth required?

​

​

​

​

• Solution

232

Transmission bandwidth of single tone FM waves

233

Generation of FM waves

• Two ways of generating FM waves
• Direct method: Generates WBFM using VCO
• Indirect method: NBFM is converted into WBFM
• Direct method
• Uses a voltage controlled oscillator (VCO) where one of the reactive elements in the tank circuit is controlled by the message signal m(t)
• In other words, the instantaneous frequency is controlled by m(t) directly
• Carrier frequency drift occurs, which is unacceptable in commercial radio service
• Can be corrected by feedback which adds to circuit complexity

​

234

Generation of FM waves

• Direct method example: Hartley oscillator

235

The capacitance C(t) consists of a fixed capacitance C₀ and voltage variable capacitor (referred to as varactor)

​

​

where k_c is variable capacitor’s sensitivity to voltage change

​

A varactor can be obtained by using a p-n junction diode biased in reverse direction

• Larger the reverse bias voltage, smaller is the variable capacitance and

vice versa

Generation of FM waves

• Direct method: Hartley oscillator (contd.)
• Its frequency depends on the tank circuit (series L₁ and L₂ parallel to C(t)) having a high-Q value

236

Generation of FM waves

• Indirect method
• Proposed by Armstrong who was also the first to recognize the noise-cleaning property of FM
• The input signal m(t) is converted into a NBFM wave with mid frequency f₁ and frequency deviation ∆f₁
• A frequency multiplier converts the NBFM signal into a WBFM signal
• Frequency multiplier is a memoryless non-linear device followed by a bandpass filter
• The output of the frequency multiplier has mid frequency of nf₁ and frequency deviation of n∆f₁

237

Generation of FM waves

• Indirect method (contd.)

238

Generation of FM waves

• Indirect method (contd.)

​

239

Generation of FM waves

• Indirect method (contd.)
• The NBFM wave is denoted by

​

​

• The output of the n^th order memoryless nonlinear device is given by

​

• The bandpass filter output is given by

240

Generation of FM waves

• Indirect method (contd.)
• The bandpass filter is designed such that it filters out the component having the mid frequency as nf_c and the bandwidth as n times the message bandwidth
• Hence, the maximum frequency deviation and mid frequency of the WBFM wave are n times the frequency deviation and mid frequency of the NBFM wave
• The multiplication factor (ratio) can be achieved by cascading several lower order nonlinear devices

241

Generation of FM waves

• Indirect method (contd.)

242

Generation of FM waves

• Indirect method – Armstrong modulator
• In a practical scenario, we may not have the relationship as desired

​

​

​

• Therefore, we go for Armstrong modulator which is wideband FM transmitter constructed using the frequency multipliers (i.e., non-linear device followed by BPF) and mixers (i.e., frequency translation using local oscillator signal)

243

Generation of FM waves

• Example 1: Armstrong modulator
• Suppose f_c1 = 200 kHz and ∆f₁=25 Hz for the NBFM wave.
• Suppose the crystal oscillator frequency f_o at the mixer stage is 10.9 MHz.
• Suppose the desired mid-frequency (f_c4) and frequency deviation (∆f₄) of the WBFM wave are 91.2 MHz and 76.8 kHz respectively.
• What are the values of n₁ and n₂ for the Armstrong modulator given in the schematic (next slide)?

244

Generation of FM waves

245

Mixer is used to ensure that the desired mid frequency is achieved for the WBFM. Frequency multipliers are used to ensure that the desired frequency deviation is achieved for the WBFM

Generation of FM waves

246

• Solution

Generation of FM waves

247

Generation of FM waves

• Example 2: Armstrong modulator
• Suppose the NBFM wave has f_c1=6 MHz and Δf₁=5 kHz. Suppose the WBFM wave has f_c=30 MHz and Δf=20 kHz. Construct the block diagram of the Armstrong modulator with square law devices, BPFs and a 6 MHz crystal oscillator. Show the block diagram. Specify the center freq. of BPFs.
• Solution

248

Generation of FM waves

• Solution (contd.)

249

Generation of FM waves

• Example 3: Armstrong modulator
• Suppose the NBFM wave has f_c1=6 MHz and Δf₁=5 kHz. Suppose the WBFM wave has f_c=30 MHz and Δf=40 kHz. Construct the block diagram of the Armstrong modulator with frequency doublers and a 9 MHz crystal oscillator. Show the block diagram. Specify the center freq. and BW of the BPF. The message bandwidth is 10 kHz
• Solution

250

Generation of FM waves

• Solution (contd.)

251

Generation of FM waves

• Solution (contd.)

252

Demodulation of FM waves

• Objective:
• To produce an output voltage that varies linearly with the instantaneous frequency of the input FM signal
• Two methods for demodulation
• Balanced frequency discriminator: direct method (i.e., no feedback)
• Phase-locked loop (PLL): indirect method (relies on frequency feedback)

253

Demodulation of FM waves

• Balanced frequency discriminator
• Operating principle: Perform FM to AM conversion and apply envelope detection to the resultant AM wave
• FM to AM conversion can be accomplished by differentiating the FM wave equation in time domain
• This is equivalent to filtering the input signal through the so called slope circuit
• The above equivalence is discussed next
• Two complementary slope circuits are used whose outputs are combined to give the envelope of the input FM signal

254

Demodulation of FM waves

• Balanced frequency discriminator
• Consider an FM wave s(t) defined by

​

​

• Upon differentiating we get the bandpass signal (AM)

​

​

• The coefficient of the sinusoid is proportional to m(t)
• Provided f_c is large (i.e., AM is not over-modulated) we can recover the envelope using an envelope detector

255

Demodulation of FM waves

• Balanced frequency discriminator
• Thus a frequency discriminator can be composed of a differentiator followed by an envelope detector
• Design of a differentiator is difficult, however we can use Fourier transform’s differentiation property

​

​

• Design of a filter whose transfer function is j2πf for all frequencies f is impractical
• Hence, an approximated filter having the following transfer function H₁(f) is considered (next slide)

256

Demodulation of FM waves

• Balanced frequency discriminator
• The filter H₁(f) acts linearly over all frequencies in the range of the transmission bandwidth B_T centered around carrier frequency f_c

​

​

• The circuit having the above transfer characteristic is called the slope circuit
• It is easier to work with lowpass equivalent signals and systems (i.e., baseband equivalent), hence they are described next

257

Demodulation of FM waves

• Balanced frequency discriminator
• Slope detector composed of tuned circuit and envelope detector is shown below
• Filter response is depicted for positive frequency side

258

Demodulation of FM waves

• Balanced frequency discriminator
• The lowpass complex envelope of the FM signal and the lowpass transfer function of the filter are given by

​

​

​

​

• The lowpass output of the filter is given by

259

Demodulation of FM waves

• Balanced frequency discriminator
• Applying inverse Fourier transform of the output gives

​

​

• Finally, the actual response (i.e., bandpass equivalent) of the slope circuit is given by

​

​

​

• The output s₁(t) exhibits both amplitude modulation and frequency modulation based on m(t)

260

Demodulation of FM waves

• Balanced frequency discriminator
• Next, we supply the slope circuit output s₁(t) to the envelope detector provided

​

​

• The envelope detector output is given by

​

​

• The constant bias term πA_cB_T/2 can be removed by implementing another slope circuit having negative slope followed by envelope detection

261

Demodulation of FM waves

• Balanced frequency discriminator
• On this basis, the second envelope detector output is given by

​

• The two envelope detector outputs are combined to remove the constant bias
• Thus the output proportional to m(t)

​

• The balanced frequency discriminator thus consists of two frequency discriminators where the slope circuit transfer functions are related as

262

Demodulation of FM waves

• Balanced frequency discriminator
• Schematic

263

Demodulation of FM waves

• Balanced frequency discriminator

264

Digital Communication

• Why digital communication?
• In analog communication, distortion and noise cannot be undone
• We can apply data compression, error correction and encryption
• Common protocol/format for storage/communication despite different types of signals: e.g., voice, image, video
• Processor and algorithms replace components and circuits
• Digital signal processing can be applied

265

Sampling theorem

• Statement of sampling theorem
• Consider a continuous time signal x(t) which is band limited (i.e., contains no frequencies higher than W Hertz)
• Then x(t) is completely determined by ordinate at a sequence of points spaced 1/2W seconds apart
• The signal x(t) can be reconstructed from its ordinate at a sequence of points spaced 1/2W seconds apart

266

Here, T_s = 1/2W represents the sampling period.

The minimum sampling rate f_s must be greater than 1/ T_s (i.e., f_s > 2W). This is referred to as the Nyquist rate

Ideal sampling process

• We begin by understanding the time domain representation of the discrete time signal x_δ(t) in terms of the continuous time signal x(t).
• Two time domain representations of the discrete time signal x_δ(t) will be considered.
• In the 1^st case, the discrete time samples are expressed in terms of the impulse train directly.
• In the 2^nd case, the discrete time samples are expressed in terms of the Fourier series representation of the impulse train.
• The Fourier transform of the x_δ(t) resulting from both time domain representations will be analyzed.
• Both provide useful insights and motivation for reconstruction of x(t)

267

Ideal sampling process

• Sampling is performed by multiplying the band-limited signal x(t) with a pulse train p(t)
• The impulse train p(t) contains impulses of magnitude placed T_s seconds apart
• The impulse train is also referred to as a Dirac comb or ideal sampling function or instantaneous sampling function
• Here T_s = 1/2W where 2W is the bandwidth of x(t)

​

​

​

​

268

Ideal sampling process

• Deriving the sampled signal x_δ(t)
• The result of sampling is a discrete-time signal consisting of sequence of samples {x(nT_s)} where n is an integer

​

​

​

​

• Alternatively, we can also use Fourier series representation of the impulse train p(t) to get an alternate expression for x_δ(t)

269

Ideal sampling process

• Deriving the sampled signal x_δ(t) (contd.)
• The impulse train p(t) is expressed as a Fourier series

​

​

where the complex Fourier coefficient p_n is given by

​

​

• Thus, we get

270

Ideal sampling process

• Deriving the sampled signal x_δ(t) (contd.)
• Multiplying the impulse train with the signal x(t), we get

​

​

​

​

• Let us look at the spectrum of the above discrete-time signal using Fourier transformation

271

Ideal sampling process

• Impact of sampling on signal spectrum
• Applying Fourier transform to x_δ(t) gives

​

272

Ideal sampling process

• Impact of sampling on signal spectrum (contd.)
• Thus, we get

​

​

​

​

​

• Periodic sampling of a signal in time domain results in a periodic spectrum in frequency domain spaced at integral multiples (+ve and –ve) of the f_s

273

Ideal sampling process

• Impact of sampling on signal spectrum (contd.)
• Note that substituting f_s = 2W and setting n = 0, we get the following simple relation

​

​

​

274

Ideal sampling process

• Sampling example 1:

275

Ideal sampling process

• Sampling example 2:

276

Ideal sampling process

• Aliasing
• Aliasing refers to the phenomenon of a high-frequency component in the spectrum of the signal seemingly taking on the identity of a lower frequency in the spectrum of its sampled version

277

Ideal sampling process

• Corrective measure to overcome aliasing
• Prior to sampling, a low-pass anti-alias filter is used to attenuate those high-frequency components of a message signal that are not essential
• The filtered signal is sampled at a rate slightly higher than the Nyquist rate.

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Ideal sampling process

• Corrective measure to overcome aliasing

279

Reconstruction from ideal sampling

• Reconstruction of x(t) from samples
• Given that the spectrum of the sampled signal x_δ(t) is periodic and non-overlapping for f_s > 2W
• We can use a lowpass filter with cut-off frequency of 2W to reconstruct the original signal x(t)

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• This can be verified by expressing the inverse Fourier transform of x_δ(t) given in slide 269
• See next slide for proof

280

Reconstruction from ideal sampling

• Reconstruction of x(t) from samples (contd.)
• Consider the discrete-time signal representation given in slide 269 (reproduced below)

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• Taking Fourier transformation of the above representation, we get an alternative representation of X_δ(f)