# ECE 445S Real-Time Digital Signal Processing Laboratory - Lecture 4 Notes

Here are a few questions that arose during the presentation of lecture 4 slides in Fall 2004. Slides numbers have been adjusted to reflect the current version of the slides.

## Ideal Lowpass Filter

Why can't an ideal (analog) lowpass filter be used in practice, e.g. for filtering a signal before sampling (slide 4-3) or reconstructing a signal from its samples (slide 4-3)?

An ideal lowpass filter has a frequency response equal to 1 over a band of low frequencies [-W, W] and 0 elsewhere. This rectangular pulse in the frequency domain has an inverse Fourier transform that is a sinc signal, which is sin(2 pi W t) / (pi W t), as shown on slide 7-10. This sinc signal is two-sided in time. It cannot be realized by a causal system.

## Difference in Sampling Theorems

Is there a difference between the Shanon Sampling Theorem and the Bandpass Sampling Theorem?

Yes. The Sampling Theorem (slide 4-6) says that the sampling rate fs must be greater than 2 fmax. The Bandpass Sampling Theorem (slide 4-12) says that the sampling rate can be lowered from fs > 2 f2 to fs > (f2 - f1) if the signal being sampled is bandpass, i.e. only has non-zero frequency content in the bands [-f2, -f1] and [f1, f2], and that sampling is followed by lowpass filtering.

## Bandpass Sampling

How does bandpass sampling work?

Bandpass sampling (slide 4-12) takes advantage of the fact that sampling replicates the spectrum of the signal being sampled at integer multiples of the sampling rate (slide 4-5).

So, if signal to be sampled has non-zero frequency content in the bands [-f2, -f1] and [f1, f2], then sampling will create an infinite but countable number of replicas. The band [f1,f2] will have replicas in the negative direction [f1 - fs, f2 - fs], [f1 - 2 fs, f2 - 2 fs], etc., and positive direction [f1 + fs, f2 + fs], [f1 + 2 fs, f2 + 2 fs], etc. Similar replicas occur for the other band [-f2, -f1].

In order to prevent aliasing,

fs > (f2 - f1)

In order for one replica of each bandpass band to be centered at the origin, the condition

fc = k fs

must hold, where k is an integer and fc is the mid-point of the frequency band [f1, f2], i.e. fc = 1/2 (f1 + f2).

Last updated 06/07/22. Send comments to bevans@ece.utexas.edu