HKN ECE 210 Exam 3 Review Session

Shomik Chatterjee, Colten Brunner, Anirudh Kumar, Alex Zhang

Logistics

• This session is recorded.
• Slides and recording can be found below about an hour after the session:
• https://wiki.illinois.edu/wiki/display/HKNDEN/HKN
• Extra drills and solutions can also be found above.

Topics

• Fourier Transform
• Signal Energy and Bandwidth
• LTI System Response with Fourier Transform
• Modulation, AM, Coherent Demodulation
• Impulse Response and Convolution
• Sampling and Analog Reconstruction
• LTIC and BIBO Stability (not on exam)

The Big Picture

• On Exam 2, the only tools we had were LTI systems and Fourier Series.

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• What did that let us do?
• Deal with any periodic signal as an input to our system and find the corresponding output.

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• But not all signals are periodic ☹

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• Introducing… the Fourier Transform!

Fourier Transform

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Note: We are engineers, so we are lazy.

Integrals are hard.

Therefore, we don’t do these integrals often, if at all.

Fascinating Fourier Fact

We only consider the Fourier Transform between time and angular frequency in this class. In physics, you can transform between ‘position’ and ‘wavenumber’ with the following transform pair:

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Transformed domains have an ‘uncertainty’ principle to them, also called the� Gabor limit, which states that:

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After some notation change, and also using p=ħk, we get a very familiar result…

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For domains not ‘scaled’ by 2π, such as time in sec and frequency in Hz, the Gabor limit has a lower bound of 1/4π.

Important Signals for Fourier Transform

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Derivative of u(t) is delta(t)

Conceptual Question

Fourier Transform Tips I

• Scaling your signal can force properties to appear; typically time delay
• Ex:

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• The properties really do matter! Take the time to acquaint yourself with them and PRACTICE. This is how you can be lazy and not do nasty integrals.

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• Remember that the Fourier Transform is linear, so you can express a spectrum as the sum of easier spectra.
• Ex: Staircase function

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• Magnitude Spectrum is even symmetric, Phase Spectrum is odd symmetric for real valued signals.

Fourier Transform Tips II

Time Travelling Tables

Find the Fourier Transform of .

Shocking Symmetries

Signal Energy and Bandwidth

LTI System Response using Fourier Transform

(Fast Fourier Transform is amazing, take ECE310 to learn more)

Modulation, AM Radio, Coherent Demodulation

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Coherent Demodulator and Envelope Detector

Superheterodyning receivers, cont.

• For envelope detection to work, we need to isolate the signal we want to receive
• Use preselector filter (before mixing) to prevent the image station problem
• Image station problem occurs when two different “station” frequencies are shifted to the same IF - is a result of mixing
• Once the station frequency has been shifted to the IF, use a sharp IF filter to receive just the station you want

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Superheterodyne Receivers

This figure was yoinked from the ECE 453 course notes - take that course if you’re interested in radio design!

Superhets and Images I

• If we didn’t have a preselector, we’d run into something called the ‘image’ problem.�
• The output of our mixer might also mix in unwanted signals, or ‘images’. This is a result of the modulation property.

Superhets and Images II

• For both up and down-conversions, there are two choices of a local oscillator.�
• Each choice of local oscillator comes with its own image frequency.�
• Below is a down-conversion from 30KHz to 20 KHz, with both LO’s shown.

Robust Radio Design

Let’s say you’re designing a radio. If you want to listen to an AM station at 1255 KHz with an IF at 455 KHz, what LO frequencies can you choose?

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Without a preselector, what would be the image station if you chose the lower LO frequency?

The Big Picture

• But sometimes frequency domain is hard.
• A triangle function is passed into a system with an impulse response of rect function. Let’s go to the frequency domain to find the output!
• The triangle becomes a sinc². The rect also becomes a sinc function.
• Now we have a sinc³ function that we need to back-convert.
• Your tables don’t have this, so it’s time to bust out the good ol’ integrals.�
• Let’s set it up:

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• If only there was another way…
• Introducing… Convolution!

Impulse Response and Convolution

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Convolution Animation #1

Convolution Animation #2

Helpful Properties for Convolution

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More Helpful Properties for Convolution

• Useful Convolution result:

Convolution is a linear operation -

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(Don’t tell the math majors)

Impulse Intuition

Given what we know about convolution, the frequency domain, and impulse properties, what must the Fourier Transform of be, and more importantly, why?

Silly Simplifications

Simplify the following expression:

Sneaky Simplifications

Simplify the following expression:

Scintillating Simplifications

Simplify the following expression:

Convoluted Convolution

Devious Derivatives

Sampling and Impulse Train (ECE310 plug)

• Let’s say we have some signal f(t), and we want to sample it, perhaps so we can store it digitally.
• The resulting sequence f[n] = f(nT), where n is an integer.
• Intuitively, sampling is simply multiplying the signal by an infinite train of impulses, spaced by T seconds apart (known as the sampling rate).

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• This is known as the A/D, or Analog to Digital conversion.
• Once we have f[n], we can use Discrete Fourier Transforms or Fast Fourier Transforms, apply digital filters, etc. etc. The possibilities are endless! Take ECE310 if you want to find out more!

Analog Reconstruction

• How do we convert from a digital sequence back to an analog sequence (D/A)?
• We want to go from here back to F(w), and then f(t).

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• How? Assuming the shifted F(w)s don’t overlap, we can multiply this sum by a rect centered at the origin, with width pi/T, and then multiply by T. Once we have F(w), we can then transform back to f(t).
• Multiplication in the frequency domain by rect is convolution in the time domain by sinc. That’s where the following seemingly magical formula comes from:

Nyquist Criterion and Why It Matters

• We made a big assumption in the previous slide! “Assuming the F(w)s don’t overlap…”
• Nyquist criterion: the condition that makes this true.
• Assuming F(w) has some bandwidth B, defined in rad/sec, then overlap won’t occur when your signal’s highest frequency spectra is half of your sampling frequency.

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• If this isn’t true, our F(w)s overlap, and we get aliasing!
• It is IMPOSSIBLE to recover the original signal when this occurs, unless you happen to know the original signal in the first place - when we apply our lowpass filter, the F(w) that comes out has been altered.
• A demonstration of this will be shown in the conceptual questions.

Nefarious Nyquist

Appalling Aliasing

The Big Picture

• When dealing with real-world systems, their properties are extremely important.
• Linearity and Time-Invariance (also known as Shift-Invariance) allow you to use the tools you’ve been learning this entire semester.
• BIBO-Stability says whether or not your system might explode if you accidentally put in the wrong input.
• Because we live in a world where time is linear, Causality indicates whether or not the system is even realizable (how can you use future inputs to calculate present inputs???)
• However if your system works off of a full set of data then you can use “future” inputs.

BIBO Stability

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Note: Systems can be BIBO stable or not. Signals are bounded or not. Signals have no notion of BIBO stability, and systems have no notion of boundedness.

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Bounded signal test: If |f(t)| ≤ α < ∞, then f(t) is bounded.

LTIC

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Pictographic Representation of Linearity

Pictographic Representation of Time-Invariance

Unpopular Unit-step

Let a system be defined by its impulse response h(t) = u(t).

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Is the system Linear? Time-Invariant? Causal? BIBO-Stable?

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If it is BIBO-Unstable, name a bounded input that will cause an unbounded output.

Ridiculous Inputs

Let a system be defined by its input-output relation y(t) = x(102841) + x(t).

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Is the system Linear? Time-Invariant? BIBO-Stable? Causal?

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If it is BIBO-Unstable, name a bounded input that will cause an unbounded output.

Ridiculous Outputs

Let a system be defined by the following input-output relation:

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Is the system Linear? Time-Invariant? BIBO-Stable? Causal?

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If it is BIBO-Unstable, name a bounded input that will cause an unbounded output.

Feedback Form

Let us know how we did and what we can do to improve!

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Completely anonymous!

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Please please please fill this out :)

Past Exam Problems

Problem 1 FA19

Problem 1 FA19

Problem 2 FA20

Problem 2 FA20

Problem 2 SP20

Note: for (b), assume α = 6

Problem 1 FA20

Problem 3 FA20

Problem 4 FA17

Problem 4 SP14

iii) Determine y(t)

Problem 3 SP 22

Problem 4 SP 20

Note: alpha is a positive integer