bevans@ece.utexas.edu
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.
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 7-10. This sinc signal is two-sided in time. It cannot be realized by a causal system.
Yes. The Sampling Theorem (slide 4-8) says that the sampling rate fs must be greater than 2 fmax. The Bandpass Sampling Theorem (slide 4-22) 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].
Bandpass sampling (slide 4-22) 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 07/16/07.
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