Multirate Digital Signal Processing Problem : process data sampled at different rates. Typical example :

1. Multirate Digital Signal Processing • Problem: process data sampled at different rates. • Typical example: • audio data sampled at 8kHz, to be played by a sound card operating at 20kHz; • Important Applications: Data Compression., Digital Communication, Spectral Estimation, Filtering. Main problem: Preserve the Frequency Spectrum resample

2. Two Main Operations: DownSampling by D: D UpSampling by I: I zeros

3. Consider an intermediate operation called Sampling. Define the sampling sequence: Then, given a sequence we define the sampled sequence .

4. We can relate the frequency spectrum of with the frequency spectrum of . Take the case of . Then we can write Modulation Modeling of Sampling

5. + = even samples add odd samples cancel

6. In general for sampling by D : We can also write in exponential form as

7. … and apply the Modulation Model as before:

8. Frequency spectrum of a sampled signal

9. Now we can determine the effect of DownSampling by an integer D. First Notice the following: D D Therefore:

11. no aliasing! aliased back

12. LPF D We need to Low Pass Filter before the decimation.

13. UpSampling by a Factor I: I Frequency Spectrum: we need to filter these out

14. LPF I

15. Multirate by a Factor I/D: LPF LPF I D combine the two LPF LPF I D