# Multirate DSP and Its Application in A/D Conversion

**This article reviews the application of multirate DSP in achieving a more efficient A/D conversion and clarifies why we need different sampling rates within a single system.**

An ADC converts a continuous-time signal,

, into a digital sequence. To this end, it samples the input signal and quantizes the amplitude of each sample.

### Periodic Sampling

The sampling operation can be mathematically modeled by first multiplying the continuous-time signal by an impulse train and then converting the result into a discrete-time sequence. The final result will be a discrete-time sequencegiven by

,

is the sampling period and its reciprocal is the sampling frequency

#####
**Figure 1.** A C/D converter multiplies the input by an impulse train s(t) and generates a discrete-time sequence. Image courtesy of Discrete-Time Signal Processing.

**Figure 1.**A C/D converter multiplies the input by an impulse train s(t) and generates a discrete-time sequence. Image courtesy of Discrete-Time Signal Processing.

Note that, in Figure 1,

is still a continuous-time signal; however,

### The Fourier Transform of a Sampled Signal

As shown in Figure 1, during sampling operation, the input is multiplied by an impulse train and we have#####
*Equation 1*

*Equation 1*

Multiplication in the time domain corresponds to convolution in the frequency domain, and we obtain

*(Appendix, Equation A1)*

#####
*Equation 2*

*Equation 2*

where

and

#####
**Figure 2. **Multiplying
a signal by an impulse train leads to replicas of the input spectrum at
multiples of the sampling frequency. Image courtesy of Discrete-Time Signal Processing.

**Figure 2.**Multiplying a signal by an impulse train leads to replicas of the input spectrum at multiples of the sampling frequency. Image courtesy of Discrete-Time Signal Processing.

### The Nyquist Sampling Theorem

We wantto be a representation of

Figure 2 suggests that we can reconstruct the original signal by applying a low-pass filter to

such that the frequency components below

To successfully reconstruct

from

### Minimum Possible Sampling Rate Requires Very Sharp Filters

Suppose that we want to sample an analog music waveform where the desired energy band is in the range(Figure 3(a)). According to the Nyquist sampling theorem, the minimum sampling frequency which can be used in this case is

### Combined Analog and Digital Filter

The obvious solution for avoiding the use of a very sharp analog filter will be using a sampling rate higher than. For example, suppose that we increase the sampling rate by a factor of

#####
**Figure 3. **(a) The spectrum of the input signal. (b) The ideal anti-aliasing filter required when using

**Figure 3.**(a) The spectrum of the input signal. (b) The ideal anti-aliasing filter required when using

#####
*.
(c) Increasing the sample rate relaxes the analog filter requirements.
(d) The overall system which uses both analog and digital filtering.
Image courtesy of IEEE.*

Fortunately, after the ADC, we have the option of using a digital filter (Figure 3(d)), which can offer both sharp transition and linear-phase response. In this way, we can sufficiently suppress the unwanted components from

to

So far, our system is not a multirate one because there is only one sampling rate used in the system. The overall system obtained from two filters (the analog prefilter and the digital filter) and the analog-to-digital converter is equivalent to that obtained by a sharp analog anti-aliasing filter with passband edge of 22kHzand an ADC sampling at 88 kHz.

But is this system efficient? Do we really need to use

samples/second to represent a signal that does not have frequency components above

Now there are two sampling rates in our system; before decimation, we were using a sampling rate of

, and after decimation, the sampling rate is

### Decimation

A discrete-time sequencethat has been downsampled by a factor of

was

#####
**Figure 4.** (a) The symbol used for factor-of-M decimation and (b) illustration of factor-of-2 decimation. Image courtesy of IEEE.

**Figure 4.**(a) The symbol used for factor-of-M decimation and (b) illustration of factor-of-2 decimation. Image courtesy of IEEE.

Since factor-of-M decimation is equivalent to sampling the underlying analog signal,

, with the sampling rate

has frequency components above

#####
**Figure 5. **(a) We need a band-limiting filter prior to decimation; (b) the filter used for the factor-of-M decimation. Image courtesy of IEEE.

**Figure 5.**(a) We need a band-limiting filter prior to decimation; (b) the filter used for the factor-of-M decimation. Image courtesy of IEEE.

### Appendix

#####
*Equation A1*

*Equation A1*