1. 2. Multirate Signals

2. Content Sampling of a continuous time signal
Downsampling of a discrete time signal
Upsampling (interpolation) of a discrete time signal

3. Sampling: Continuous Time to Discrete Time
Time Domain:
Frequency Domain:

4. Reason:
same
same

5. Antialiasing Filter sampled noise noise
For large SNR, the noise can be aliased,
… but we need to keep it away from the signal

6. Example 1. Signal with Bandwidth 2. Sampling Frequency
Anti-aliasing Filter
1. Signal with Bandwidth
2. Sampling Frequency
3. Attenuation in the Stopband
Filter Order:
slope

7. Downsampling: Discrete Time to Discrete Time
Keep only one every N samples:

8. Effect of Downsampling on the Sampling Frequency
The effect is resampling the signal at a lower sampling rate.

9. Effect of Downsampling on the Frequency Spectrum
We can look at this as a continuous time signal sampled at two different sampling frequencies:

10. Effect of Downsampling on DTFT
Y(f) can be represented as the following sum (take N=3 for example):

11. Effect of Downsampling on DTFT
Since we obtain:

12. Downsampling with no Aliasing
If bandwidth then
Stretch!

13. Antialiasing Filter
In order to avoid aliasing we need to filter before sampling:
LPF
LPF
noise
aliased

14. Example Let be a signal with bandwidth sampled at Then Passband:
LPF
Let be a signal with bandwidth
sampled at
Then Passband:
Stopband:
LPF

15. See the Filter: Freq. Response…
h=firpm(20,[0,1/22, 9/44, 1/2]*2, [1,1,0,0]);
passband
stopband 2f

16. … and Impulse Response

17. Upsampling: Discrete Time to Discrete Time
it is like inserting N-1 zeros between samples

18. Effect of Upsampling on the DTFT
“ghost” freq.
“ghost” freq.
it “squeezes” the DTFT
Reason:

19. Interpolation by Upsampling and LPF

20. SUMMARY:
LPF
LPF
LPF
LPF