systems: definition s a system is a transformation from an input signal into an output signal....
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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
Systems and Properties: Linearity
Linearity:
S][][][ 2211 nxanxanx ][][][ 2211 nyanyany
][1 nx ][1 ny
][2 nx ][2 nyS
S
Systems and Properties: Time Invariance
if
then
][nx ][nyS
S
][ Dnx ][ Dny
D D
time
time
time
time
Systems and Properties: Stability
S][nx ][ny
Bounded InputBounded Output
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
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:
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.
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][
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][
Frequency Response of an FIR Filter
njenx 0][ njeHny 00][ Filter
N
n
njenhH0
][)(
is the Frequency Response of the Filter
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 )(
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][
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:
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
)(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
)(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
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
Why this is Important
][nx ][nyFilter
Recall from the DTFT
deXnx nj)(2
1][
Then the output
deHXny nj)()(2
1][
Which Implies )()(][)( XHnyDTFTY
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
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.
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.
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)