 # Fundamentals of Multirate Systems

Gordana Jovanovic Dolecek (Institute Inaoe, Mexico)
DOI: 10.4018/978-1-60566-026-4.ch253
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## Abstract

Digital signal processing (DSP) is an area of science and engineering that has been rapidly developed over the past years. This rapid development is a result of significant advances in digital computers technology and integrated circuits fabrication (Mitra, 2005; Smith, 2002). Classical digital signal processing structures belong to the class of single-rate systems since the sampling rates at all points of the system are the same. The process of converting a signal from a given rate to a different rate is called sampling rate conversion. Systems that employ multiple sampling rates in the processing of digital signals are called multirate digital signal processing systems. Sample rate conversion is one of the main operations in a multirate system (Harris, 2004; Stearns, 2002).
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## Background

### Decimation

The reduction of a sampling rate is called decimation, because the original sample set is reduced (decimated). Decimation consists of two stages: filtering and downsampling, as shown in Figure 1. The discrete input signal is u(n) and the signal after filtering is x(n). Both signals have the same input sampling rate fi.

Downsampling reduces the input sampling rate fi by an integer factor M, which is known as a downsampling factor. Thus, the output discrete signal y(m) has the sampling rate fi /M. It is customary to use a box with a down-pointing arrow, followed by a downsampling factor as a symbol to represent downsampling, as shown in Figure 2.

The output signal y(m) is called a downsampled signal and is obtained by taking only every M-th sample of the input signal and discarding all others,

y(m)=x(mM).(1)

The operation of downsampling is not invertible because it requires setting some of the samples to zero. In other words, we can not recover x(n) from y(m) exactly, but can only compute an approximate value.

In spectral domain downsampling introduces the repeated replicas of the original spectrum at every 2π/M. If the original signal is not bandlimited to π/M, the replicas will overlap. This overlapping effect is called aliasing. In order to avoid aliasing, it is necessary to limit the spectrum of the signal before downsampling to below π/M. This is why a lowpass digital filter (from Figure 1) precedes the downsampler. This filter is called a decimation or antialiasing filter.

Three useful identities summarize the important properties associated with downsampling (Jovanovic Dolecek, 2002). The First identity states that the sum of the scaled, individually downsampled signals is the same as the downsampled sum of these signals. This property follows directly from the principle of the superposition (linearity of operation). The Second identity establishes that a delay of M samples before the downsampler is equivalent to a delay of one sample after the downsampler, where M is the downsampling factor. The Third identity states that the filtering by the expanded filter followed by downsampling, is equivalent to having downsampling first, followed by the filtering with the original filter, where the expanded filter is obtained by replacing each delay of the original filter with M delays. In the time domain this is equivalent to inserting M-1 zeros between the consecutive samples of the impulse response.

The polyphase decimation, which utilizes polyphase components of a decimation filter, is a preferred structure for decimation, because it enables filtering to be performed at a lower sampling rate (Diniz, da Silva & Netto, 2002).

## Key Terms in this Chapter

Software Radio: The term software radio generally refers to a radio that derives its flexibility through software while using the same piece of hardware, that is, the same piece of hardware can perform different functions at different times. In that way a completely reconfigurable physical hardware is obtained by programming in software.

Continuous-Time Signals: Continuous signals are defined along a continuum of time t and thus are represented by continuous independent variables, for example x c (t). Continuous-time signals are often referred to as analog signals.

Discrete-Time Signals: Discrete-time signals are defined at discrete time values and thus the independent variable has discrete values n , as for example x(n) .

Digital Filter: The filter is a discrete-time system, which changes the characteristics of the input discrete signal in a desired manner to obtain the discrete output signal.

Digital Filter Bank: Set of digital bandpass filters with the common input or a common output.

Sampling: The generation of a discrete-time signal x ( n ) from a continuous signal x c ( t ) is called sampling, where x(n)= x c (nT). T is called the sampling period and its inverse 1/ T is the sampling frequency or the sampling rate.

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