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

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

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

See the k-th channel:

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

See the k-th channel:

In fact:

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

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

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

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

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

Therefore the modulator becomes: substitute for this…

IFFT

Use Noble Identity: IFFT UNBUFFER

Similarly This is noncausal!!! We need a time delay:

Write all filters in matrix form This matrix yields the FFT

Therefore the Demodulator: Substitute for the vector …

In block diagram: FFT

Use Noble Identity: FFT BUFFER

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

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:

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:

Modulator: each filter has order 200/8=25 8-point IFFT UNBUFFER

Demodulator: 8-point FFT BUFFER