Linear time-invariant systems operate at a single sampling rate i.e. the sampling rate is the same at the input and at the output of the system, and at all the nodes inside the system. Thus, in an LTI system, the sampling rate doesn’t change in different stages of the system. Systems that use different sampling rates at different stages are called the multirate systems. The multirate techniques are used to convert the given sampling rate to the desired sampling rate, and to provide different sampling rates through the system without destroying the signal components of interest. In this chapter, we consider the sampling rate alterations when changing the sampling rate by an integer factor. We describe the basic sampling rate alteration operations, and the effects of those operations on the spectrum of the signal.
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Linear time-invariant systems operate at a single sampling rate i.e. the sampling rate is the same at the input and at the output of the system, and at all the nodes inside the system. Thus, in an LTI system, the sampling rate doesn’t change in different stages of the system. Systems that use different sampling rates at different stages are called the multirate systems. The multirate techniques are used to convert the given sampling rate to the desired sampling rate, and to provide different sampling rates through the system without destroying the signal components of interest.
In this chapter, we consider the sampling rate alterations when changing the sampling rate by an integer factor. We describe the basic sampling rate alteration operations, and the effects of those operations on the spectrum of the signal.
TIME-DOMAIN REPRESENTATION OF DOWN-SAMPLING AND UP-SAMPLING
Converting the sampling rate means that one discrete signal is converted into another discrete signal with a different sampling rate. Two discrete signals with different sampling rates can be used to convey the same information. For example, a bandlimited continuous signal xc(t) might be represented by two different discrete signals {x[n]} and {y[n]} obtained by the uniform sampling of the original signal xc(t) with two different sampling frequencies FT and FT’
And
(1) where
T= 1/
FT and
T’=1/
FT’ are the corresponding sampling intervals. When the sampling frequencies
FT and
FT’ are chosen in such a way that each of them exceeds at least two times the highest frequency in the spectrum of
xc(
t), the original signal
xc(
t) can be reconstructed from either {
x[
n]} or {
y[
n]}. Hence, the two signals operating at two different sampling rates are carrying the same information. By using the discrete-time operations, signal {
x[
n]} can be converted to {
y[
n]}, or vice versa, with minimal signal distortions.
The basic operations in sampling rate alteration process are the sampling rate decrease and the sampling rate increase. Employing two operators can perform the sampling rate alteration: a down-sampler for the sampling rate decrease, and an up-sampler for the sampling rate increase. The down-sampler and the up-sampler are the sampling rate alteration devices since they decrease or increase the sampling rate of the input sequence.