1. Multi Carrier Modulation and Channelizers
We want to transmit a number of signals over the same channel TX
DAC RX
Channel

2. Need for Multirate M signals at samples/sec Baseband channel
1 signal at samples/sec
Since the data rate cannot decrease (we do not want to loose information), we need to constrain

3. Modulator (without carrier)
Channels: 1 2 k
M-1

4. See the k-th channel:

5. Demodulator (without carrier)
Same for the demodulator
Channels: k
M-1 1 2

6. See the k-th channel:

7. In fact:

8. Efficient Implementation of the Filters
Choose all filters in the modulator/demodulator from the same prototype:
with real
prototype filter

9. First notice the following:1.
has transfer function 2.
has transfer function

10. Extend it to all the filters:
in the z domain set to obtain:
Similarly:

11. All these filters and are nicely related to the polyphase decomposition of the prototype filter
Then:

12. Write all these terms in vector form:
This matrix yields M x IFFT

13. Therefore the modulator becomes:
substitute for this…

14. IFFT

15. Use Noble Identity:
IFFT
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16. Similarly
This is noncausal!!!
We need a time delay:

17. Write all filters in matrix form
This matrix yields the FFT

18. Therefore the Demodulator:
Substitute for the vector …

19. In block diagram:
FFT

20. Use Noble Identity:
FFT
BUFFER

21. 8 channels
50 dB attenuation between channels
80% useful bandwidth
Example:
Prototype Filter

22. Step 1: design the prototype filter:
Transition band:
Estimated order:
… too conservative!
Design the filter:
h=firpm(199, [0, 1/20, 1/16, 1/2]*2, [1,1,0,0]);
Frequency Response:

23. Step 2: Polyphase decomposition of the prototype filter.
with
The impulse responses of the M polyphase filters are computed by reshaping the impulse response of the filter into M rows:

24. Modulator: each filter has order 200/8=25
8-point IFFT
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25. Demodulator:
8-point FFT
BUFFER