# A Summary of Multirate Digital Signal Processing

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• Topic: Digital signal processing, Filter, Signal processing
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The Hong Kong Polytechnic University Department of Electronic and Information Engineering EIE413 Digital Signal Processing Individual Essay

Multirate Digital Signal Processing

Instructor: W.C. Siu By HAN Shilu 07828567D

EIE413 Digital Signal Processing

Multirate Digital Signal Processing

Abstract
In traditional digital signal processing system, there is always only one simple sampling rate (that is, the sampling frequency). The output signal has the same sampling rate with the input. In modern digital systems, however, there is an

increasing need to process data at more than one sampling rate. Sometimes the output of the system is required to have a different sampling rate of the input signal. This has lead the development of multirate digital signal processing, which is a new sub-area in DSP. For example, the sampling rate for an audio CD (compact disc) is 44.1 kHz. If we want to transfer data from the CD to a DAT (digital audio tape) at a sampling rate of 48 kHz, we need to increase the frequency of the data first using a multirate approach. There are two primary options we have in multirate processing. The first is decimation. The sampling rate fs of a given signal x[n] is decreased. This approach is also called down sampling. The second is interpolation. We can increase the sampling rate fs of the given signal x[n]. This approach is also called up sampling. In this paragraph, we are going to discuss the general principles of decimation and interpolation in multirate processing and sampling conversion by non-integer factor.

Decimation by an Integer Factor
Decimation is simply the process of decreasing the sampling frequency of an input signal to a desired value. In this part, we shall confine our attention to a decrease by an integer factor M. For example, we have an input signal I = {1, 3, 5, 7, 9, 11, 13, 2, 4, 6, 8, 10, 12}. The output signal y[n] is formed by taking every Mth sample of the input signal. If M=4, we should just take every fourth sample of x[n] to obtain the desired signal y[m] = {1, 9, 4, 12}. According to Nyquist sampling theorem, we know that aliasing will occur in the down sampled signal because of the reduced sampling rate. After the signal is down sampled by a factor M, the new sampling frequency fsM becomes fs/M, where fs is the original sampling frequency. The folding frequency after 1

EIE413 Digital Signal Processing

Multirate Digital Signal Processing

down sampling becomes fs/2M. If the original signal has frequency components than the new folding frequency, aliasing will be introduced to the output signal. In order to solve the problem, it is necessary to process the original signal x[n] through a lowpass filter H(z) which has a cut-off frequency of fs/2M. The corresponding normalized cut-off frequency should be vc = (fs/2M) / fs = 1/2M. The filtered output in terms of z-transform can be written as W(z) = H(z)X(z). Where X(z) is the z-transform of the original signal x[n]. After passing through the lowpass filter, the value of the down sampled signal y(m) can be obtained from the filtered output: y(m) = w(mM). The process of down sampling by a factor of 3 is shown in Figure 1A. And Figure 1B shows the corresponding spectral for x(n), w(n) and y(m).

Figure 1A. Down sampling with M=3.

Figure 1B. Spectral after down sampling.

It has to be decided whether IIR or FIR should be used for the lowpass filtering required. The use of an IIR filter has an obvious short coming. Previous outputs are required to compute the Mth output. The computation is too complicated. We cannot take the advantage that we only have to compute every Mth output. If, however, an FIR filter is used, we can do our computations at the rate of fs /M. We can obtain the desired output y(m) by the equation: y(m) = w(mM) = ∑ .

Another advantage of an FIR filter is that it is easy to design a linear phase filter 2

EIE413 Digital Signal Processing which is desirable in many cases....