2. Multirate Signals.

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Presentation transcript:

2. Multirate Signals

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

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

Reason: same same

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

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

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

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

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

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

Effect of Downsampling on DTFT Since we obtain:

Downsampling with no Aliasing If bandwidth then Stretch!

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

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

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

… and Impulse Response

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

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

Interpolation by Upsampling and LPF

SUMMARY: LPF LPF LPF LPF