HKN ECE 310 Final Exam Review Session - PowerPoint Presentation

Slide1

HKN ECE 310 Final Exam Review Session

Corey Snyder

Ben Eng

Slide2

Topics

LSIC Systems and BIBO Stability

Impulse Response and Convolution

Z-TransformDTFT and Frequency Response

Sampling

Discrete Fourier Transform (DFT)

Fast Fourier Transform (FFT)

Circular Convolution

Digital Filter Design

Rate Conversion and Multirate Systems

D/A Conversion

Slide3

LSIC Systems

Linearity

Satisfy Homogeneity and Additivity

Can be summarized by SuperpositionIf

and

then

Shift InvarianceIfthen andCausalityOutput cannot depend on future input values

 

Slide4

BIBO Stability

Three ways to check for BIBO Stability:

Pole-Zero Plot (more on this later)

Absolute summability of the impulse responseGiven

then the system is BIBO stable

A bounded input

yields a bounded output Ex: vs. Absolute Summability 

Slide5

Impulse Response

]

is the impulse response

System output to an

input

Convolution in the time/sample domain is multiplication in the transformed domain, both the z-domain and frequency domain.

 

Slide6

Convolution

System must be:

Linear

Shift Invariant

Popularly done graphically

Can also be done algebraically

 

Slide7

Z-Transform

We mainly focus on the one-sided, or unilateral, z-transform

Typically perform inverse z-transform by inspection or by Partial Fraction Decomposition

Important properties:

Multiplication by n:

Delay Property #1:

Make sure to note the Region of Convergence (ROC) for your transforms!More in the next slide!DTFT is only defined if the ROC contains the unit circle 

Slide8

BIBO Stability Revisited

Pole-Zero Plot

For an LSI system: if the ROC contains the unit circle, this system is BIBO stable

The ROC is anything greater than the outermost pole if the system/signal is causal

The ROC is anything less than the innermost pole if the system/signal is anti-causal

If we sum multiple signals, the ROC is the

intersection of each signal’s ROCWhat if the ROC is or ? This is marginally stable, but unstable for ECE 310 purposesFor unstable systems, you are commonly asked to find a bounded input that yields an unbounded output. Few ways to do this:Pick an input that excites the poles of the system.If the system’s impulse response is not absolutely summable, will work frequently works too, like when is unbounded, e.g.  

Slide9

Discrete Time Fourier Transform

Important Properties:

Periodicity!

Linearity

Symmetries (Magnitude, angle, real part, imaginary part)

Time shift and modulation

Product of signals and convolutionParseval’s RelationKnow your geometric series sums! 

Slide10

Frequency Response

For any stable LSI system:

What is the physical interpretation of this?

The DTFT is simply the z-transform evaluated along the unit circle!

It makes sense that the system must be stable and LSI since the ROC will contain the unit circle, thus ensuring that the DTFT is well defined

Why is the frequency response nice to use in addition to the z-transform?

is an eigenfunction of LSI systemsBy extension:  

Slide11

Magnitude and Phase Response

Very similar to ECE 210

Frequency response, and all DTFTs for that matter, are

periodic

Magnitude response is fairly straightforward

Take the magnitude of the frequency response, remembering that

= 1For phase response:Phase is “contained” in termsRemember that cosine and sine introduce sign changes in the phaseLimit your domain from to .For real-valued systems:Magnitude response is even-symmetricPhase response is odd-symmetric 

Slide12

Ideal A/D Conversion

Sampling via an impulse train will yield infinitely many copies of the analog spectrum in the digital frequency domain

 

Slide13

Discrete Fourier Transform

What is the relationship between the DTFT and the DFT?

 

Slide14

DFT Properties

Circular

shift

Circular modulationCircular convolutionWe must amend our DTFT properties with the “circular” term because the DFT is defined over a finite length signal and assumes periodic extension of that finite signal.

Slide15

Zero-PaddingWe can improve the resolution of the DFT simply by adding zeros to the end of the signal

This doesn’t change the frequency content of the DTFT!

Instead, it increases the number of samples the DFT takes of the DTFT

This can be used to improve spectral resolution

Slide16

Fast Fourier Transform

Computational efficient implementation of the DFT

Ordinary DFT requires N

2 multiplies and N(N – 1) addsFFT requires only

computations

Two main forms of FFT

Decimation in TimeDecimation in Frequency (Not covered in this class)Use Butterfly Structures to represent multiplies and adds 

Slide17

Decimation in Time

Divide sequence into two groups

Before we continue, remember that

We can form the first half of the DFT from these two sequences

And the second half…

Continually halve the sequences until you reach size 2

 

Slide18

Fast Linear Convolution

In order to obtain system response, we can multiply DFTs and take inverse DFT

Be careful, this is not the same as convolution in time, but rather cyclic convolution in time

Therefore, in order to perform linear convolution from DFTs, we must first zero pad signals in order to make wrap-around terms go to zeroIf

has length

and

has length , then the resulting convolution is of length .Pad with zerosPad with zerosThis will allow multiplication of FFTs to produce linear convolution result from cyclic convolutionNote: we also typically do extra padding so that the signals are of a length that is a power of the FFT’s radix. 

Slide19

Linear Phase Filters

FIR vs. IIR – is there feedback?

Types of FIR Filters:

Type 1Odd Length and Even SymmetricType 2Even Length and Even SymmetricType 3Odd Length and Odd SymmetricType 4

Even Length and Odd Symmetric

Slide20

Linear Phase Filter Design

Why linear phase?

Satisfy causality and filter should have finite number of terms

Generalized Linear Phase vs. Linear PhaseBegin with ideal magnitude response,

, for a length N filter

Introduce linear phase

Perform inverse DTFT of to obtain Window with window function  

Slide21

Digital Filter Structure

Suppose we have the LCCDE:

we can express the system in two different

Direct Forms

.

 

Direct Form IDirect Form II

Slide22

Tips for Linear Phase Filter Design

Perform the IDTFT integral for high-pass filters from

.

This will allow you to do one integral instead of two

Only design low-pass filters!

Why?

EasierHow?Use modulation property (multiplication by cosine) to obtain other filters from low-pass filters 

Slide23

Upsampling

If we upsample by

, we will interpolate

zeros between each sample

if

mod

; and else What happens in the frequency domain?Think about what happens when we oversample a signal, i.e. above NyquistWhat does the frequency response look like after upsampling?Shrink x-axis by factor of What should be in order to obtain a desirable frequency response?Remove extra copies and correct amplitude for conservation of energy 

Slide24

Downsampling

If we downsample by

, we keep every

sample (decimate the rest)

What happens in the frequency domain?

Frequency response stretches by a factor of

Amplitude reduces by a factor of (think conservation of energy)Anti-aliasing filter prevents downsampling from aliasing our signal 

Slide25

Ideal D/A

Want

, but we want to only take one copy of the DTFT

Thus, we should low-pass filter from

(domain of the central copy of the DTFT)

Remember that

Remember that multiplication in the frequency domain is convolution in the time domainThus, , where is obtained by multiplying y[n] by an impulse train 

Slide26

Realizable D/A: Zero-Order Hold

Ideal D/A is not practical because generating delta impulses is not achievable

Zero-Order Hold (ZOH) gives us a suitable approximation to the Ideal D/A

The ZOH multiplies each sample by a rectangular pulse of width T (our sampling rate)Thus,

where

is the rectangular pulse provided by the ZOH

is an analog filter that corrects the distortion presented by the ZOH  

Slide27

How does the ZOH Change our Spectrum?

Slide28

Upsampled D/A

Upsampling prior to D/A conversion can make recovery simpler

i.e. Compensator

can be simpler to implement

Don’t forget that we filter after upsampling!

Full system is then:

[LPF][ZOH]Upsampling effectively increases our sampling frequency, thus our ZOH pulse can be narrower and give us a better staircase approximationThis ‘smoother staircase’ will be easier to rectify with the compensatori.e. the transition bandwidth will be largerIn the frequency domain, we see the frequency axis compress by ; however, the analog frequencies upon recovery do not change! 

Slide29

P1: CTFT to DTFT

The continuous time Fourier Transform (CTFT) of

is given as:

between

and

andDetermine andforand 

Slide30

P2 : Non-unique Digital Frequencies

A

cosine is sampled at

. The digital frequency that it is mapped to is

.

Which of the following

sampling frequencies could cause this?a.) b.) c.) c.)  

Slide31

P3: DFT Matrices and the DFT

Draw the

DFT matrix

Given a length 5 sequence

, and length 8 sequence

,which is just a zero padded

What are the common terms in the DFT of these two sequences? 

Slide32

P4: DFT Shift Property

Given length 5 sequence

with corresponding transform

. Determine the DFT of

.

 

Slide33

P5: Circular Convolution

Determine the circular convolution of sequences

. What is the equivalent matrix operation?

 

Slide34

P6 : La Mariposa (The Butterfly)

Slide35

Butterfly Structure Exercise

Draw the butterfly structure of a length 8 Decimation in Time FFT for

 

Slide36

P7: Zero Padding

Given sequences

. To what length must they be zero-padded to such that their linear convolution is the same as the cyclic convolution?

Suppose there was a radix-3 FFT algorithm; to what length would both of these sequences need to be zero-padded to?

 

Slide37

P8: FIR and IIR Systems

Can you guess which of these systems are FIR and IIR?

 

Slide38

P9: Filter Structure

Draw the Direct-Form II structure for the following difference equation:

 

Slide39

P10 : GLP Filters

Determine the Filter Type for the following systems:

 

Slide40

P11 : Window Method

Design a length 34

HPF with a Hamming

Window and cutoff frequency

.

Please also include in your solution a side of fries and a

small soft drink.  

Slide41

P12 : Up, Down, and Around

is a bandlimited sequence to

It′s Fourier Transform is given as

Given sampling frequency of

Determine

,

The is ideal with cutoff and magnitude 1.  

Slide42

P13 : ADC /DSP /DAC/WTF

,

.