systems: definition s a system is a transformation from an input signal into an output signal....
TRANSCRIPT
![Page 1: Systems: Definition S A system is a transformation from an input signal into an output signal. Example: a filter Filter SIGNAL NOISE](https://reader035.vdocument.in/reader035/viewer/2022062321/56649da25503460f94a8ec17/html5/thumbnails/1.jpg)
Systems: Definition
][nx ][ny
S
A system is a transformation from an input signal into an output signal .
][nx ][ny
Example: a filter
][ns
][nv
][nx ][][ nsny Filter
SIGNAL
NOISE
![Page 2: Systems: Definition S A system is a transformation from an input signal into an output signal. Example: a filter Filter SIGNAL NOISE](https://reader035.vdocument.in/reader035/viewer/2022062321/56649da25503460f94a8ec17/html5/thumbnails/2.jpg)
Systems and Properties: Linearity
Linearity:
S][][][ 2211 nxanxanx ][][][ 2211 nyanyany
][1 nx ][1 ny
][2 nx ][2 nyS
S
![Page 3: Systems: Definition S A system is a transformation from an input signal into an output signal. Example: a filter Filter SIGNAL NOISE](https://reader035.vdocument.in/reader035/viewer/2022062321/56649da25503460f94a8ec17/html5/thumbnails/3.jpg)
Systems and Properties: Time Invariance
if
then
][nx ][nyS
S
][ Dnx ][ Dny
D D
time
time
time
time
![Page 4: Systems: Definition S A system is a transformation from an input signal into an output signal. Example: a filter Filter SIGNAL NOISE](https://reader035.vdocument.in/reader035/viewer/2022062321/56649da25503460f94a8ec17/html5/thumbnails/4.jpg)
Systems and Properties: Stability
S][nx ][ny
Bounded InputBounded Output
![Page 5: Systems: Definition S A system is a transformation from an input signal into an output signal. Example: a filter Filter SIGNAL NOISE](https://reader035.vdocument.in/reader035/viewer/2022062321/56649da25503460f94a8ec17/html5/thumbnails/5.jpg)
Systems and Properties: Causality
the effect comes after the cause.
Examples:
S][nx ][ny
]3[4]2[2]1[3][ nxnxnxny Causal
]3[4]2[2]1[3][ nxnxnxny Non Causal
![Page 6: Systems: Definition S A system is a transformation from an input signal into an output signal. Example: a filter Filter SIGNAL NOISE](https://reader035.vdocument.in/reader035/viewer/2022062321/56649da25503460f94a8ec17/html5/thumbnails/6.jpg)
Finite Impulse Response (FIR) Filters
][nx ][nyFilter
N
nxhnxnhny0
][][][*][][
][][...]1[]1[][]0[][ NnxNhnxhnxhny
Filter Coefficients
General response of a Linear Filter is Convolution:
Written more explicitly:
![Page 7: Systems: Definition S A system is a transformation from an input signal into an output signal. Example: a filter Filter SIGNAL NOISE](https://reader035.vdocument.in/reader035/viewer/2022062321/56649da25503460f94a8ec17/html5/thumbnails/7.jpg)
Example: Simple Averaging
][nx ][nyFilter
]9[...]1[][10
1][ nxnxnxny
Each sample of the output is the average of the last ten samples of the input.
It reduces the effect of noise by averaging.
![Page 8: Systems: Definition S A system is a transformation from an input signal into an output signal. Example: a filter Filter SIGNAL NOISE](https://reader035.vdocument.in/reader035/viewer/2022062321/56649da25503460f94a8ec17/html5/thumbnails/8.jpg)
FIR Filter Response to an Exponential
njenx 0][ njeHny 00][ Filter
njN
jN
nj eehehny 000
0
)
0
)( ][][][
Let the input be a complex exponential
Then the output is
njenx 0][
![Page 9: Systems: Definition S A system is a transformation from an input signal into an output signal. Example: a filter Filter SIGNAL NOISE](https://reader035.vdocument.in/reader035/viewer/2022062321/56649da25503460f94a8ec17/html5/thumbnails/9.jpg)
Example
njenx 0][ njeHny 00][ Filter
Consider the filter
]9[...]1[][10
1][ nxnxnxny
with inputnjenx 1.0][
Then 4137.11.0
101.09
0
1.0 6392.01
1
10
1
10
11.0 j
j
jj e
e
eeH
and the output njj eeny 1.04137.16392.0][
![Page 10: Systems: Definition S A system is a transformation from an input signal into an output signal. Example: a filter Filter SIGNAL NOISE](https://reader035.vdocument.in/reader035/viewer/2022062321/56649da25503460f94a8ec17/html5/thumbnails/10.jpg)
Frequency Response of an FIR Filter
njenx 0][ njeHny 00][ Filter
N
n
njenhH0
][)(
is the Frequency Response of the Filter
![Page 11: Systems: Definition S A system is a transformation from an input signal into an output signal. Example: a filter Filter SIGNAL NOISE](https://reader035.vdocument.in/reader035/viewer/2022062321/56649da25503460f94a8ec17/html5/thumbnails/11.jpg)
Significance of the Frequency Response
k
njk
keXnx ][Filter
If the input signal is a sum of complex exponentials…
k
njk
keYny ][
… the output is a sum is a sum of complex exponential.
Each coefficient is multiplied by the corresponding frequency response:
kX kkk XHY )(
![Page 12: Systems: Definition S A system is a transformation from an input signal into an output signal. Example: a filter Filter SIGNAL NOISE](https://reader035.vdocument.in/reader035/viewer/2022062321/56649da25503460f94a8ec17/html5/thumbnails/12.jpg)
Example
Consider the Filter
Filter][nx ][ny
]4[...]1[][5
1][ nxnxnxnydefined as
Let the input be:
)7.03.0cos(2)2.01.0cos(3][ nnnx
Expand in terms of complex exponentials:
njjnjj
njjnjj
eeee
eeeenx
3.07.03.07.0
1.02.01.02.0
0.10.1
5.15.1][
![Page 13: Systems: Definition S A system is a transformation from an input signal into an output signal. Example: a filter Filter SIGNAL NOISE](https://reader035.vdocument.in/reader035/viewer/2022062321/56649da25503460f94a8ec17/html5/thumbnails/13.jpg)
Example (continued)
The frequency response of the filter is (use geometric sum)
j
jjj
e
eeeH
1
1
5
1...1
5
1)(
54
njjnjj
njjnjj
eeHeeH
eeHeeHny
3.07.03.07.0
1.02.01.02.0
0.12.00.12.0
5.11.05.11.0][
Then
2566.12566.1
6283.06283.0
647.0)1.0(,647.0)2.0(
904.0)1.0(,904.0)1.0(jj
jj
eHeH
eHeH
with
)956.13.0cos(294.1)428.01.0cos(712.2][ nnny Just do the algebra to obtain:
![Page 14: Systems: Definition S A system is a transformation from an input signal into an output signal. Example: a filter Filter SIGNAL NOISE](https://reader035.vdocument.in/reader035/viewer/2022062321/56649da25503460f94a8ec17/html5/thumbnails/14.jpg)
The Discrete Time Fourier Transform (DTFT)
Given a signal of infinite duration with
define the DTFT and the Inverse DTFT
][nx n
n
njenxnxDTFTX ][][)(
deXXIDTFTnx nj)(2
1)(][
Periodic with period 2
)()2( XX
![Page 15: Systems: Definition S A system is a transformation from an input signal into an output signal. Example: a filter Filter SIGNAL NOISE](https://reader035.vdocument.in/reader035/viewer/2022062321/56649da25503460f94a8ec17/html5/thumbnails/15.jpg)
)(rad
|)(| X
0
)(X
)(HzF2SF
2SF 0
If the signal is real, then ][nx )()( * XX
General Frequency Spectrum for a Discrete Time Signal
Since is periodic we consider only the frequencies in the interval
![Page 16: Systems: Definition S A system is a transformation from an input signal into an output signal. Example: a filter Filter SIGNAL NOISE](https://reader035.vdocument.in/reader035/viewer/2022062321/56649da25503460f94a8ec17/html5/thumbnails/16.jpg)
)(1
1)(
1
0
Nj
NjN
n
nj We
eeX
Then
Example: DTFT of a rectangular pulse …
Consider a rectangular pulse of length N
0 1N
1
][nx
n
2/sin
2/sin )( 2/)1(
N
eW NjN
where
![Page 17: Systems: Definition S A system is a transformation from an input signal into an output signal. Example: a filter Filter SIGNAL NOISE](https://reader035.vdocument.in/reader035/viewer/2022062321/56649da25503460f94a8ec17/html5/thumbnails/17.jpg)
0 1N
1
][nx
n
DTFT
Example of DTFT (continued)
-3 -2 -1 0 1 2 30
2
4
6
8
10
12
( )NW
N
2
N
2
N
![Page 18: Systems: Definition S A system is a transformation from an input signal into an output signal. Example: a filter Filter SIGNAL NOISE](https://reader035.vdocument.in/reader035/viewer/2022062321/56649da25503460f94a8ec17/html5/thumbnails/18.jpg)
Why this is Important
][nx ][nyFilter
Recall from the DTFT
deXnx nj)(2
1][
Then the output
deHXny nj)()(2
1][
Which Implies )()(][)( XHnyDTFTY
![Page 19: Systems: Definition S A system is a transformation from an input signal into an output signal. Example: a filter Filter SIGNAL NOISE](https://reader035.vdocument.in/reader035/viewer/2022062321/56649da25503460f94a8ec17/html5/thumbnails/19.jpg)
Summary Linear FIR Filter and Freq. Resp.
][nx ][nyFilter
1
0
][][][*][][N
nxhnxnhny
Filter Definition:
Frequency Response: ,][)(1
0
N
n
njenhH
DTFT of output )()()( XHY
![Page 20: Systems: Definition S A system is a transformation from an input signal into an output signal. Example: a filter Filter SIGNAL NOISE](https://reader035.vdocument.in/reader035/viewer/2022062321/56649da25503460f94a8ec17/html5/thumbnails/20.jpg)
Frequency Response of the Filter
][nx ][nyFilter
Frequency Response:
,][)(1
0
N
n
njenhH
We can plot it as magnitude and phase. Usually the magnitude is in dB’s and the phase in radians.
![Page 21: Systems: Definition S A system is a transformation from an input signal into an output signal. Example: a filter Filter SIGNAL NOISE](https://reader035.vdocument.in/reader035/viewer/2022062321/56649da25503460f94a8ec17/html5/thumbnails/21.jpg)
Example of Frequency Response
Again consider FIR Filter
The impulse response can be represented as a vector of length 10
]9[...]1[][10
1][ nxnxnxny
1.0...1.0,1.0h
Then use “freqz” in matlab
freqz(h,1)
to obtain the plot of magnitude and phase.
![Page 22: Systems: Definition S A system is a transformation from an input signal into an output signal. Example: a filter Filter SIGNAL NOISE](https://reader035.vdocument.in/reader035/viewer/2022062321/56649da25503460f94a8ec17/html5/thumbnails/22.jpg)
Example of Frequency Response (continued)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-200
-100
0
100
Normalized Frequency ( rad/sample)
Pha
se (
degr
ees)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-80
-60
-40
-20
0
Normalized Frequency ( rad/sample)
Mag
nitu
de (
dB)