M. Lustig, EECS UC Berkeley

EE123Digital Signal Processing

Lecture 16

M. Lustig, EECS UC Berkeley

Lab II - Time-Frequency

• compute spectrograms with w/o windowing

M. Lustig, EECS UC Berkeley

Lab II - Time-Frequency

• Compute with overlapping window

M. Lustig, EECS UC Berkeley

Lab II - Time-Frequency

• Look at temporal/frequency resolution tradeoffs:

M. Lustig, EECS UC Berkeley

FM Broadcast Radio - KPFA 94.1MHz

M. Lustig, EECS UC Berkeley

Spectrogram of Broadcast FMNon-demodulated

FM demodulated

M. Lustig, EECS UC Berkeley

Spectrogram of Broadcast FMFM demodulated

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

M. Lustig, EECS UC Berkeley

Demodulate subcarriers: Example 92KHz

demodulate by 92KHz

M. Lustig, EECS UC Berkeley

Demodulate subcarriers: Example 92KHz

• Filter and decimate

• FM demodulate and filter

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

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

=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!

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!

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!

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:

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=-∞,...,∞

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

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

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

⇡/Mx[n]

x̃[n] x̃d[n] = x̃[nM ]

M. Lustig, EECS UC Berkeley

Anti-Aliasing

LPF↓M

⇡�⇡

Xd

X

⇡�⇡

M=3