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����PRESENTATION ON �DISCRETE TIME SIGNAL

  • BRANCH-E & TC ENGG
  • SUBJECT- DIGITAL SIGNAL PROCESSING
  • CHAPTER – 2 – DISCRETE TIME SIGNAL & SYSTEM
  • TOPIC- DISCRETE TIME SIGNAL
  • SEM-6TH
  • FACULTY – Er. ARADHANA DAS (Sr. LECTURER E & TC ENGG DEPARTMENT)
  • AY-2021-2022, SUMMER-2022

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2.0 Introduction

  • Signal: something conveys information, represented mathematically as functions of one or more independent variables. Classified as:
  • Continuous-time (analog) signals, discrete-time signals, digital signals
  • Signal-processing systems are classified along the same lines as signals: Continuous-time (analog) systems, discrete-time systems, digital systems

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2.1 Discrete-Time Signals: Sequences

  • Discrete-Time signals are represented as

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  • In sampling of an analog signal xa(t):

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  • 1/T (reciprocal of T) : sampling frequency

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Cumbersome, so just use

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Figure 2.1 Graphical representation of a discrete-time signal

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Abscissa: continuous line

: is defined only at discrete instants

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Figure 2.2

EXAMPLE

Sampling the analog waveform

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Basic Sequence Operations

  • Sum of two sequences

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  • Product of two sequences

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  • Multiplication of a sequence by a number α

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  • Delay (shift) of a sequence

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Basic sequences

  • Unit sample sequence (discrete-time impulse, impulse, Unit impulse)

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  • 离散时间单位脉冲(样本)序列, 区别连续时间单位冲激函数(continuous-time unit impulse function δ(t) )。

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Basic sequences

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  • arbitrary sequence

A sum of scaled, delayed impulses

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Basic sequences

  • Unit step sequence

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First backward difference

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Basic Sequences

  • Exponential sequences

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  • A and α are real: x[n] is real
  • A is positive and 0<α<1, x[n] is positive and decrease with increasing n
  • -1<α<0, x[n] alternate in sign, but decrease in magnitude with increasing n
  • : x[n] grows in magnitude as n increases

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Combining Basic sequences

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  • If we want an exponential sequences that is zero for n <0, then

Cumbersome

simpler

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Basic sequences

  • Sinusoidal sequence

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Exponential Sequences

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Complex Exponential Sequences

Exponentially weighted sinusoids

Exponentially growing envelope

Exponentially decreasing envelope

is refered to

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difference between continuous-time and discrete-time complex exponentials or sinusoids

  • : frequency of the complex sinusoid or complex exponential
  • : phase

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Periodic Sequences

  • A periodic sequence with integer period N

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EX. 2.1 Examples of Periodic Sequences

  • Suppose it is periodic sequence with period N

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  • Suppose it is periodic sequence with period N

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EX. 2.1 Examples of Periodic Sequences

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EX. 2.1 Non-Periodic Sequences

  • Suppose it is periodic sequence with period N

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High and Low Frequencies in Discrete-time signal

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(b) w0 = π/8 or 15π/8

(c) w0 = π/4 or 7π/4

(d) w0 = π

Frequency: The rate at which a repeating event occurs.

(a) w0 = 0 or 2π

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