eece 301 note set 26 dtft system analysis
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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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7/31/2019 EECE 301 Note Set 26 DTFT System Analysis
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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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7/31/2019 EECE 301 Note Set 26 DTFT System 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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7/31/2019 EECE 301 Note Set 26 DTFT System Analysis
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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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7/31/2019 EECE 301 Note Set 26 DTFT System Analysis
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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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7/31/2019 EECE 301 Note Set 26 DTFT System Analysis
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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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7/31/2019 EECE 301 Note Set 26 DTFT System Analysis
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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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7/31/2019 EECE 301 Note Set 26 DTFT System Analysis
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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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7/31/2019 EECE 301 Note Set 26 DTFT System Analysis
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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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7/31/2019 EECE 301 Note Set 26 DTFT System Analysis
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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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7/31/2019 EECE 301 Note Set 26 DTFT System Analysis
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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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7/31/2019 EECE 301 Note Set 26 DTFT System Analysis
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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
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