eece 301 note set 26 dtft system analysis

7/31/2019 EECE 301 Note Set 26 DTFT System Analysis

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EECE 301

Signals & Systems

Prof. Mark Fowler

Note Set #26

D-T Systems: DTFT Analysis of DT Systems Reading Assignment: Sections 5.5 & 5.6 of Kamen and Heck


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Ch. 1 Intro

C-T Signal Model

Functions on Real Line

D-T Signal Model

Functions on Integers

System Properties

LTICausal

Etc

Ch. 2 Diff EqsC-T System Model

Differential Equations

D-T Signal ModelDifference Equations

Zero-State Response

Zero-Input Response

Characteristic Eq.

Ch. 2 Convolution

C-T System Model

Convolution Integral

D-T System Model

Convolution Sum

Ch. 3: CT FourierSignalModels

Fourier Series

Periodic Signals

Fourier Transform (CTFT)

Non-Periodic Signals

New System Model

New Signal

Models

Ch. 5: CT FourierSystem Models

Frequency Response

Based on Fourier Transform

New System Model

Ch. 4: DT Fourier

SignalModels

DTFT

(for Hand Analysis)DFT & FFT

(for Computer Analysis)

New Signal

Model

Powerful

Analysis Tool

Ch. 6 & 8: LaplaceModels for CT

Signals & Systems

Transfer Function

New System Model

Ch. 7: Z Trans.

Models for DT

Signals & Systems

Transfer Function

New System

Model

Ch. 5: DT Fourier

System Models

Freq. Response for DT

Based on DTFT

New System Model

Course Flow DiagramThe arrows here show conceptual flow between ideas. Note the parallel structure between

the pink blocks (C-T Freq. Analysis) and the blue blocks (D-T Freq. Analysis).


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5.5: System analysis via DTFT

][nh][nx

= =

=

iinxih

nxnhny

][][

][][][

Recall that in Ch. 5 we saw how to use frequency domain methods to analyze

the input-output relationship for the C-T case.

We now do a similar thing for D-T

Define the Frequency Response of the D-T system

We now return to Ch. 5

for its DT coverage!

Back in Ch. 2, we saw that a D-T system in zero state has an output-input

relation of:

=

=n

njenhH ][)(

DTFT ofh[n]

Perfectly parallel to the

same idea for CT systems!!!


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From Table of DTFT properties: )()(][][ HXnhnx

So we have:

)(

][

H

nh

)(

][

X

nx ][][][ nxnhny =

)()()( = HXY

)()()(

)()()(

+=

=

HXY

HXYSo

Soin general we see that the system frequency response re-shapes the input

DTFTs magnitude and phase.

System can:

-emphasize some frequencies

-de-emphasize other frequencies

Perfectly parallel to the same

ideas for CT systems!!!

The above shows how to use DTFT to do general DT system analyses

virtually all of your insight from the CT case carries over!


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Now lets look at the special case: Response to Sinusoidal Input

,...3,2,1,0,1,2,3)cos(][ 0 =+= nnAnx

From DTFT Table:

[ ]


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So what does Y( ) look like?

[ ]


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So

))(cos()(][ 000 ++= HnAHny

)(H)cos( 0 + nA ))(cos()( 000 ++ HnAH

System changes amplitude and phase of

sinusoidal input

Perfectly parallel to the same

ideas for CT systems!!!


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Im

Re

Example 5.8 (error in book)

Suppose you have a system described by+= jeH 1)(

+=

+= nnnnx

2sin2)0cos(2

2sin22][

And you put the following signal into it

Cosine with = 0

Find the output.

So we need to know the systems frequency response at only 2 frequencies.

212

j

eH

+=

Im

Rej1

21)0( 0 =+= jeH

421

j

ej

==


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=

+

=

===

42sin22

22sin2

2][

422)0cos(2)0(][

2

1

nHnHny

nHny

Since the system is linear we can consider each of the input

terms separately.

And then add them to get the complete response

+=

42

sin224][

nny


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Note: In the above example we used

)(H)sin( 0 + nA ))(sin()( 000 ++ HnAH

Q: Why does that follow?

A: It is a special case of the cosine result that is easy to see:

- convert sin(0n + ) into a cosine form

- apply the cosine result

- convert cosine output back into sine form

which is the sine version of our result above for a cosine input


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Analysis of Ideal D-T lowpass Filter (LPF)

2 3 44 3 2

filterlowpassIdeal)( =H

1

-B B

Just as in the CT case we can specify filters. We looked at the ideal

lowpass filter for the CT case here we look at it for the DT case.

Cut-off frequency = B rad/sample

As always with DT we only need to look here

This slide shows how a DT filter might be employed but ideal filters cant


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D-T Ideal

LPFDAC

ADC @

Fs = 1/T

F~

2Fs@sample

~2

~let

=

= FB

This slide shows how a DT filter might be employed but ideal filters can t

be built in practice. Well see later a few practical DT filters.

x(t) x[n] y[n] y(t)


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We know the frequency response of the ideal LPF so find its impulse response:

From DTFT Table :

= n

BBnh

sinc][

Why cant an ideal LPF exist in practice??

][nh

n

Key Point: h[n] is non-zero here

starts before the impulse that makes it is even applied!

Cant build an Ideal LPF

(Same thing is true in C-T)

Causal Lowpass Filter (But not Ideal)


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Causal Lowpass Filter (But not Ideal)

In practice, the best we can do is try to approximate the ideal LPF

If you go on to study DSP youll learn how to design filters that do a good

job at this approximation

Here well look at two seat of the pants approaches to get a good LPF

Approach #1 Truncate & Shift Ideal h[n]

= 2][

Nnhnh truncapprox

][nhapprox

n

ShiftTruncate

][nhtrunc

n

=)(,0

22],[

][

evenNotherwise

Nn

Nnh

nhideal

trunc

N+1 non-zerosamples

Lets see how well these work


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-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

/

|H(

)|

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

/

|H()|

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

/

|H()|

N = 20

N = 60

N = 120

Some general insight: Longer lengths for the truncated impulse response

Gives better approximation to the ideal filter response!!

Lets see how well these work

Frequency Response

of a filter truncated to

21 samples

Frequency Response

of a filter truncated to

61 samples

Frequency Response

of a filter truncated to121 samples

Approach #2: Moving Average Filters


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Approach #2: Moving Average Filters

Here is a very simple, low quality LPF:

=

=

otherwise

nnh

,0

1,0,2

1

][

][nh

2

1

-3 -2 -1 1 2 3 4 5

n

To see how well this works as a lowpass filter we find its frequency response:

[ ]

=

+=

+==

j

jj

n

nj

e

eeenhH

12

1

2

1

2

1][)( 10

By definition of

the DTFT

Only 2 non-zero

terms in the sum


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[ ] ( )

)cos(12

1)cos(1

2

2

)(sin)(cos)cos(212

1

)sin())cos(1()(

1

22

2

212

21

+=+=

+++=

++=

=

H

Now, to see what this looks like we find its magnitude.

Euler!It is now in rect. form

Trig. ID

Now.. Plot this to see if it is agood LPF!

[ ]

[ ])sin())cos(1(2

1

121)(

+=

+=

j

eH j

H l f hi fil f i d


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We could try longer moving average filters:

=

=otherwise

Nn

Nnh

,0

1,,2,1,0,1

]["

Heres a plot of this filters freq. resp. magnitude:

2/ 2/

repeats

periodically

repeats

periodically

)(H

1

2

1

Wellthis does attenuate high frequencies but doesnt really stop them!

It is a low pass filter but not a very good one!

How do we make a better LPF???

LowFrequencies HighFrequenciesHighFrequencies

1


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Plots of various Moving Average Filters

=

=

otherwise

NnNnh

,0

1,,2,1,0,1

]["

N= 2

N= 3

N= 5

N= 10

N= 20

We see that increasing the length of the all-ones moving average

filter causes the passband to get narrower but the quality of the filter

doesnt get better so we generally need other types of filters.


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The two approaches to DT filters weve seen here are simplistic approaches

There are now very powerful methods for designing REALLY good DT

filters well look at some of these later in this course.

A complete study of such issues must be left to a senior-level course in DSP!!

Comments on these Filter Design Approaches

eece 301 note set 26 dtft system analysis

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