2-Multirate Signals
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Transcript 2-Multirate Signals
2. Multirate Signals
Content
• Sampling of a continuous time signal
• Downsampling of a discrete time signal
• Upsampling (interpolation) of a discrete time signal
Sampling: Continuous Time to Discrete Time
Time Domain:
x[n] x(nTs )
x(t )
n
t
Fs
Frequency Domain:
X DTFT ( f )
X FT (F )
F
Fs Fs / 2
1
2
X DTFT ( f ) Fs
X F kF
k
FT
s
F f Fs
Fs / 2
1
2
Fs
F
f
F
Fs
Reason:
x[n] (t nT )
x(t )
s
n
(t nT )
s
n
FT x(t ) (t nTs ) X FT ( F ) * Fs ( F nFs ) Fs X FT ( F nFs )
n
n
n
same
FT x[n] (t nTs )
n
same
x[n]e
n
j 2n
F
Fs
F
X DTFT
Fs
Antialiasing Filter
x(t )
Anti-aliasing
Filter
X FT (F )
x[n] x(nTs )
H (F )
Fs
H (F )
noise
B
Fs
2
sampled
noise
Fs
Fs
F
FPASS B
Fs
2
B
Fs
FSTOP Fs B
2
For large SNR, the noise can be aliased,
… but we need to keep it away from the signal
Fs
F
Fs B
Example
x[n] x(nTs )
H (F )
x(t )
Anti-aliasing
Filter
1. Signal with Bandwidth
2. Sampling Frequency
B 3 kHz
Fs 11kHz
3. Attenuation in the Stopband
H (F )
dB
0
Fs
A 60 dB
B
Fs B
3
8
F (logscale)
Filter Order:
60
slope
60
20N
log10 83
N 7
Downsampling: Discrete Time to Discrete Time
y[m] x[m N]
x[n]
N
1 2
3 2 1
6
3
4
5
0
y[m] x[m N]
7
2
n
1
0
Keep only one
every N samples:
1 2
3 2 1
6
3
0
N 3
4
5
7
n
1
m
Effect of Downsampling on the Sampling Frequency
y[m] x[m N]
x[n]
N
y[m] x[m N]
Fs 2 Fs1 / N
Ts1
3 2 1
6
3
0
Ts1
4
5
7
t
0
Ts 2
t
Ts 2 N Ts1
The effect is resampling the signal at a lower sampling rate.
Effect of Downsampling on the Frequency Spectrum
y[m] x[m N]
x[n]
N
X( f )
Fs 2 Fs1 / N
Fs1
Fs1
1
0
Fs1
1
Y( f ) ?
F
f
We can look at this as a continuous time signal sampled at two different sampling
frequencies:
X( f )
Fs1
x[n] s(nTs1 )
s (t )
S (F )
Fs1
1
Fs1
1
F
f
Fs1
0
1
Y( f )
F
y[m] s(mTs 2 )
0
Fs 2
Fs 2
Fs1
Fs 2
1
0
Fs 2
1
Fs1
F
f
Effect of Downsampling on DTFT
Y(f) can be represented as the following sum (take N=3 for
example):
F
Y( f ) Y
F
s2
1 F
X
3 Fs1
Fs 2
Fs1
Fs 2
1
0
Fs 2
1
0
Fs 2
1
1
F
f
Fs 2
Fs1
1
F
f
1 F 2 Fs 2
X
3 Fs1
Fs 2
Fs1
Fs1
1 F Fs 2
X
3 Fs1
Fs 2
Fs1
Fs 2
0
Fs 2
1
Fs1
F
f
Effect of Downsampling on DTFT
F 1 N 1 F kFs 2
Y
X
F
N
F
k 0
s1
s2
Since Fs 2 Fs1 / N we obtain:
1 N 1 f k
Y f X
N k 0 N N
X( f )
x[n]
y[m] x[m N]
N
Fs 2 Fs1 / N
Downsampling with no Aliasing
y[m] x[m N]
x[n]
N
Fs 2 Fs1 / N
Fs1
If bandwidth
b
Xf
then Y ( f )
1
2N
1
N
X
f
N
B
B
Nb
Fs 2 Fs1 / N
Y f
1
1
N
1
2N
1
2N
12
b
1
2
f
Nb
12
Stretch!
b
B
Fs1
1
2
f
Antialiasing Filter
In order to avoid aliasing we need to filter before sampling:
f PASS B / Fs1 b
1
f STOP Fs 2 B / Fs1
2N
LPF
X( f )
b
1
N
y[m]
LPF
noise
x[n]
b
1
2N
N
Fs1
Fs 2 Fs1 / N
f
b
Nb
1
2N
f1
aliased
1
2
f2
Example
y[m]
LPF
4
x[n]
Fs1
Fs 2 Fs1 / 4
Let x[n] be a signal with bandwidth B 2kHz
sampled at Fs 44kHz
Then Passband:
f PASS
B
2
1
Fs1 44 22
Stopband:
f STOP
Fs 2 B 1 1
9
Fs1
4 22 44
f PASS b
LPF
f STOP
f
1
N
b
See the Filter: Freq. Response…
h=firpm(20,[0,1/22, 9/44, 1/2]*2, [1,1,0,0]);
passband
stopband
Magnitude (dB)
50
0
-50
-100
-150
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Normalized Frequency ( rad/sample)
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Normalized Frequency ( rad/sample)
0.9
1
Phase (degrees)
0
-200
-400
-600
-800
2f
… and Impulse Response
h[n]
0.3
0.25
0.2
0.15
0.1
0.05
0
-0.05
0
5
10
15
20
25
Upsampling: Discrete Time to Discrete Time
x[n]
y[m]
N
Fs1
x[n]
1
0
x[k ] [m kN ]
k
Fs 2 NFs1
N
n
0
m
it is like inserting N-1
zeros between samples
Effect of Upsampling on the DTFT
x[n]
y[m]
N
Fs 2 NFs1
Fs1
“ghost” freq.
X( f )
Y ( f ) X ( Nf )
Y( f )
“ghost” freq.
1
2
b 12
1
2
f
1
2 1
N N
b 1 2
N N N
1
2
it “squeezes” the DTFT
Reason:
Y( f )
y[m]e
m
j 2fm
j 2fnN
x
[
n
]
e
X ( Nf )
n
f
Interpolation by Upsampling and LPF
f STOP 1Nb
LPF
f2
X( f )
x[n]
N
Fs1
1
2
b
1
2
LPF
Fs 2 NFs1
f PASS
b
N
Y( f )
y[m]
f
1
2
1
2
1
N
b
N
1
N
1
2
f
b
N
1
2
SUMMARY:
f PASS b B / Fs1
f STOP N1 b Fs 2 B / Fs1
LPF
X( f )
Y( f )
y[m]
x[n]
b
LPF
N
Fs1
Nb
f
1
2N
f STOP 1Nb Fs1 B / Fs 2
LPF
f2
X( f )
f PASS
x[n]
N
Fs1
1
2
b
1
2
f
f
1
2
b
N
B / Fs 2
Y( f )
LPF
y[m]
1
2
b
N
1
2
f