EE 345S 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.

Ideal Lowpass Filter

An ideal lowpass filter has a frequency response equal to 1 over a band of low frequencies [-W, W] and 0 elsewhere. From EE 313, 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 7-8. This sinc signal is two-sided in time. It cannot be realized by a causal system.

Difference in Sampling Theorems

Yes. The Shannon Sampling Theorem (slide 4-6) says that the sampling rate f_s must be greater than 2 fmax. The Bandpass Sampling Theorem (4-9) says that the sampling rate can be lowered from f_s > 2 f₂ to f_s > (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].

Bandpass Sampling

Bandpass sampling (slide 4-9) takes advantage of the fact that sampling replicates the spectrum of the signal being sampled at integer multiples of the sampling rate (slide 4-3).

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 - f_s, f₂ - f_s], [f₁ - 2 f_s, f₂ - 2 f_s], etc., and positive direction [f₁ + f_s, f₂ + f_s], [f₁ + 2 f_s, f₂ + 2 f_s], etc. Similar replicas occur for the other band [-f₂, -f₁].

In order to prevent aliasing,

f_s > (f₂ - f₁)

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

f_c = k f_s

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

Last updated 09/15/04. Send comments to bevans@ece.utexas.edu