Ch6 Fourier

On board, did demonstration to show what convolution looks like for two sin curves of the same vs different frequencies, and of different phases.

Discrete-time filters in continuous frequency space

Back to your textbook, in Chapter 6:

Thus for fixed \(m\) and arbitrary \(n\), we have

Gain

\(\left|y_{m, n}\right| \leq\left|H\left(-m \omega T_{s}\right)\right|\left|x_{m, n}\right|\)

So we define gain or attenuation of this transformation as \(G(\zeta)=|H(\zeta)|\)

. . .

We define phase rotation as \(\Theta(\zeta)=\theta\), where \(H(\zeta)=|H(\zeta)| e^{i \theta}\)

. . .

The filter \(h\) will attenuate the signal at frequencies where \(|H(\zeta)|<1\) and amplify the signal at frequencies where \(|H(\zeta)|>1\). It will phase shift the signal by \(\Theta(\zeta)\).

High-pass and low-pass filters

The FIR filter \(\mathbf{h}=\left\{h_{k}\right\}_{k=0}^{L}\) with discrete time Fourier transform \(H(\zeta)\) is a lowpass filter if \(|H(0)|=1\) and \(|H(\pi)|=0\); \(\mathbf{h}\) is a highpass filter if \(|H(0)|=0\) and \(|H(\pi)|=1\).