0.6 Sampling: review  (Page 2/2)

Nyquist Shannon sampling theorem :

Assume that the continuous-time signal $x\left(t\right)$ is bandlimited , meaning there is some $B>0$ such that $X\left(f\right)=\mathcal{F}\left\{x,\left(,t,\right)\right\}\left(f\right)=0$ whenever $\left|f\right|>B$ .

If this signal $x$ is sampled uniformly at a sufficient rate, namely at $f_e>2B$ , then $x\left(t\right)$ can be mathematically reconstructed perfectly from only those discrete samples.

Reconstruction (Ideal low pass filtering)

If the sample rate has been high enough though, namely

then, the Fourier transform of the original signal can be recovered by cutting off the periodic copies:

where $f_c$ has to be chosen such that $B<f_c<f_e-B$ (see [link] right). (The first copy to be cut off lies over the interval $[f_e-B,f_e+B]$ .) A possible choice is $f_c=f_e/2$ .

In the time domain, this corresponds to convolving the sampled signal with the ideal low-pass filter ( $\text{sinc}$ ). More precisely: Translating [link] into the time domain, then using the linearity of the convolution and [link] , we find the ideal low-pass filtering (reconstruction) formula:

Note that the pre-factor $2\tau f_c=2f_c/f_e$ takes into account the mismatch between sampling rate $f_e$ and the cut-off frequency $f_c$ . When choosing $f_c=f_e/2$ , it disappears (becomes 1).

Practical reconstruction : Recall that the sinc filter is not realizable since it requires using all samples of the past and thefuture. Since a bandlimited signal can not be time-limited at the same time (Heisenberg), this would in theory lead to infinitely manyoperations.

If $f_e>>2B$ , then a filter different from the ideal sinc-filter can be used. In fact, the reconstruction $x\left(t\right)=x_e\left(t\right)*b\left(t\right)$ is valid for any low-pass filter $b$ (not only for $b\left(t\right)=2f_c\text{sinc}\left(2f_{c}t\right)$ ) as long as we assure that the spectrum of the filter $b$ is equal to 1 for all frequencies in $[-B,B]$ and zero for frequencies larger $f_e-B$ . (Note that we do not specify the phase). Usually, one designs a filter such that

1. The filter's spectrum equals 1 in an interval $[-f_p,f_p]$ called passing band . It must contain the signal's frequency-band $[-B,B]$ .
2. The filter attenuates all frequencies beyond $f_e-f_p$ .
3. The transition band of the filter is $[f_p,f_e-f_p]$ . This range of frequencies can be chosen aslarge as $[B,f_e-B]$ by setting $f_p=B$ and as short as desired, but always containing the point $f_e/2$ by setting $f_p$ smaller but as close to $f_e/2$ as desired.

All this assures that we keep the entire spectral information of $x\left(t\right)$ contained in $[-B,B]$ , yet remove all spectral copies (see [link] ). Sampling an ideal reconstruction filter is a method to a design a low pass filter. Here, point 2 in the above list helps avoidingaliasing in such a filter. However, the resulting filters are not ideal as FIR filters possible since not all samples can be used, resultingin Gibbs-like phenomena (see [link] on the right).

Matlab uses the command fir1 to compute finite impulse response filters (FIR). A larger transition band allows for shorter filters to be used, thusspeeding up computation. [link] shows a length 61 FIR filter with ideal cutoff frequency at $f_c=1/8=0.125$ . Its transition band is roughly $[0.10,0.15]$ . As is common in digital filtering, thesampling frequency is assumed to be $f_e=1$ .

Power and Energy from Sampling

We summarize a few facts.

Good quality when above Nyquist We are now in a position to make more precise how many samples are enough to well approximate the power of a signal via [link] , or its energy per time over a large interval via [link] , depending on the context. Roughly, the signal must be sampled at least at Nyquist rateso that the samples and their DFT faithfully represent the signal and its Fourier transform. Clearly, sampling at a much higher rate than Nyquist will improve the approximation.

Power independent of sampling rate We conclude that changing the sampling rate will not change power, as long as we stay above Nyquist, and at least approximately. Before we study how to change the sampling rate in thenext sections, we give a quick demonstration of [link] using a simple band-limited signal: a filter.

Ideal filter and FIR filter The energy of the ideal filter with cutoff frequency $f_c$ is $\left|\right|2f_c\text{sinc}\left(2f_{c}t\right){\left|\right|}^2=2·f_c$ which follows easily from the power spectrum being 1 on an interval of length $2·f_c$ and zero else, using Plancherel. To compute the power of a digital low pass FIR filter $b=(b_0,...,b_{K-1})$ of length $K$ with approximate cutoff frequency $f_c$ we study its DFT $\hat{b}$ : we note that ${\hat{b}}_k=1$ for roughly $2f_{c}K$ of the indices $k$ and ${\hat{b}}_k=0$ for the other indices. Note that $f_e=1$ and $0<f_c<1/2$ for a digital filter. Using [link] we find

This fits perfectly with [link] since for a digital filter $f_e=1$ and, thus, $K=L$ . The approximation becomes better, the sharper the transition band of the filter $b$ , i.e., the longer the filter is.

Compare again with [link] for a concrete filter with length $K=61$ , $f_c=1/8=0.125$ and power $P_b=0.{003}^{\text{'}}891$ .