ee123 digital signal processingee123/sp14/notes/... · digital signal processing lecture 16. m....
TRANSCRIPT
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M. Lustig, EECS UC Berkeley
EE123Digital Signal Processing
Lecture 16
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M. Lustig, EECS UC Berkeley
Lab II - Time-Frequency
• compute spectrograms with w/o windowing
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M. Lustig, EECS UC Berkeley
Lab II - Time-Frequency
• Compute with overlapping window
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M. Lustig, EECS UC Berkeley
Lab II - Time-Frequency
• Look at temporal/frequency resolution tradeoffs:
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M. Lustig, EECS UC Berkeley
FM Broadcast Radio - KPFA 94.1MHz
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M. Lustig, EECS UC Berkeley
Spectrogram of Broadcast FMNon-demodulated
FM demodulated
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M. Lustig, EECS UC Berkeley
Spectrogram of Broadcast FMFM demodulated
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M. Lustig, EECS UC Berkeley
Filter Mono and down
after filtering
after decimation
• To play we need to filter the right signal• Downsample to 48KHz so we can play on the computer
120KHz
24KHz
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M. Lustig, EECS UC Berkeley
Demodulate subcarriers: Example 92KHz
demodulate by 92KHz
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M. Lustig, EECS UC Berkeley
Demodulate subcarriers: Example 92KHz
• Filter and decimate
• FM demodulate and filter
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M. Lustig, EECS UC Berkeley
Topics
• Last time– Ideal reconstruction D/C– D.T processing of C.T signals– C.T processing of D.T signals (ha?????)
• Today– Changing Sampling Rate via DSP– Downsampling– Upsampling
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H(ej!) = e�j!�
Hc(j⌦) = e�j⌦�T
M. Lustig, EECS UC Berkeley
Example:
• What is the time-domain operation when Δ is not an integer (Δ=1/2)?
Non-integer delay:
xc(t) yc(t) y[n]x[n]D/C Hc(jΩ) C/D
T T
Let: delay of ΔT in time
C.T recon delay ΔT sampling
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=X
k
x[k]sinc(n� k ��)
y[n] = yc(nT ) = xc(nT � T�)
=X
k
x[k]sinc
✓t� kT � T�
T
◆�����t=nT
M. Lustig, EECS UC Berkeley
Example: Non Integer Delay
• The block diagram is only for interpretation!
yc(t) = xc(t��)
T’s cancel!
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h[n] = sinc(n��)
M. Lustig, EECS UC Berkeley
Example: Non Integer Delay
Example: a discrete delta is a representation of a sampled sinc
shifted by partial samples results in many coefficients!
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M. Lustig, EECS UC Berkeley
DownSampling
• Much like C/D conversion• Expect similar effects:
–Aliasing–mitigate by antialiasing filter
• Finely sampled signal ⇒ almost continuous–Downsample in that case is like sampling!
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x[n]xd[n] = x[nM ]
= xc(nT ) = xc(nMT|{z})T 0
X(ej!) =1
T
X
k
Xc
j
!
T|{z}� 2⇡
T|{z}k
!!
Xd(ej!) =
1
MT
X
k
Xc
✓j
✓!
MT� 2⇡
MTk
◆◆⌦ ⌦s
M. Lustig, EECS UC Berkeley
Changing Sampling-rate via D.T Processing
MDownsampling:
The DTFT:
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M. Lustig, EECS UC Berkeley
Changing Sampling-rate via D.T Processing
X(ej!) =1
T
X
k
Xc
j
!
T|{z}� 2⇡
T|{z}k
!!
Xd(ej!) =
1
MT
X
k
Xc
✓j
✓!
MT� 2⇡
MTk
◆◆⌦ ⌦s
The DTFT:
we would like to bypass Xc and go from X(ejω)⇒Xd (ejω)
two counterse.g., k: hours, i: minutes
substitute r = kM + i i=0,1,...,M-1k=-∞,...,∞
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Xd(ej!) =
1
MT
X
k
Xc
✓j
✓!
MT� 2⇡
MTk
◆◆
=1
M
M�1X
i=0
1
T
1X
�1Xc
✓j
✓!
MT� 2⇡
MTi� 2⇡
Tk
◆◆
| {z }X(ej(
!M �
2⇡M i))
Xd(ej!) =
1
M
M�1X
i=0
X(ej(!M �
2⇡M i))
M. Lustig, EECS UC Berkeley
Changing Sampling-rate via D.T Processing
stretchby M
replicate
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Xd(ej!) =
1
M
X
i=0
X⇣ej(w/M�2⇡i/M)
⌘
⇡�⇡
Xd
X
M. Lustig, EECS UC Berkeley
Changing Sampling-rate via D.T Processing
⇡�⇡
M=2
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Xd(ej!) =
1
M
X
i=0
X⇣ej(w/M�2⇡i/M)
⌘
⇡�⇡
Xd
X
M. Lustig, EECS UC Berkeley
Changing Sampling-rate via D.T Processing
⇡�⇡
M=3
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⇡/Mx[n]
x̃[n] x̃d[n] = x̃[nM ]
M. Lustig, EECS UC Berkeley
Anti-Aliasing
LPF↓M
⇡�⇡
Xd
X
⇡�⇡
M=3