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DSP Final Project

Multi-rate in Digital Signal Processing

14/01/04

Khalaf Batiha

Multirate in Digital Signal Processing

I. Introduction

Multi-rate signal processing studies digital signal processing systems which

include sample rate conversion. Multi-rate signal processing techniques are

necessary for systems with different input and output sample rates, but may

also be used to implement systems with equal input and output rates.

Multi-rate signal processing changes the sampling rate of the discrete time

signal. Multi-rate signal processing can be used in digital systems to reduce the

complexity of computations and to reduce the transmission data rate; the

input signal is downsampled before processing. And also, it’s changed when

two systems connected to each other and they use different Sampling rates.

Hence the conversion is needed to match these systems.

The digital signal processing (DSP) systems can be classified into two types;

namely, single rate systems in which all data are sampled at fixed sample rate,

and multi-rate systems in which the sampling rate can be changed.

II. Sampling Rate Conversion

Sampling rate conversion is converting the signal rate from one rate to

another. If we form discrete sequence from sampling analog signal with

period , then we can form ,

which is analogous to sampling analog signal with period , By

Sampling conversion. The Sampling Theorem must be taken in consideration to

ensure the reconstruction of original signal; changing the information carried by

the signal as little as possible, or in other words minimizes the discrepancy.

and its samples

clarifies an example of an analog signal with its samples at deferent

sampling frequencies (i.e. deferent data rates), the deferent data rates can be

achieved by changing the sampling rate wich can be done in two methods;

convert to analogue, and then re-sample. “ ”, and

changing sampling rate in digital domain . We will discuss the

Multi-rate method; in which we could convert the sampling rate into two modes

of operation; decimation (for downsampling), and interpolation (for Upsampling).

i. Decimation by factor M

Downsampling is the process in which the sampling rate is decreasing

by an integer factor ,

It follows

Where x[n] is the input of the system, y[n] is the system output, Fs is

the sampling frequency,F’s the output sampling freqquency, and M is

the downsampling factor. As indicated in fig.2.

Fig.2. and its down-sampling version

Fig.2. indicates an example of the down-sampling for some signal by the

factor .

We note that the down Sampling removes samples, hence decreases the

sampling rate. But we should take in our consideration the aliasing problem; to

prevent the aliasing in this case; we use the digital anti-aliasing filter (low pass) to

remove frequencies higher than some pre-determined threshold:

So, the digital frequency is,

Remark: down sampling combined with the digital filter is called

Decimation.

Filtering operation is Linear and Time Invariant, but down sampling is not,

however, decimation is not linear time invariant operation.

ii. Interpolation (Upsampling)

Upsampling process is the process of increasing the sampling rate by factor

(Integer), such that the new sampling frequency is equal to the

original sampling frequency multiplied by the factor as follow:

Interpolation can be achieved by Zero-filling by adding zeros

between successive samples, or Low pass filtering, in which the frequency of

the LPF

And,

Fig.3. indicates an upsampling conversion example for a given input signal with upsampling factor , where is the interpolated sequence of

Fig. 3. The input signal vs. the interpolation version

There is a combined case obtained from the two aforementioned methods

called non-integer ratio conversion; it is done in two steps; firstly do the

interpolation by factor , and secondly the decimation by , this results into

non-integer ration conversion ( ). The block diagram in Fig.4 indicates this

case.

Fig. 3. The flow diagram of the non-integer sampling conversion

Please find the attached Matlab code in the compressed file which does the interpolation and

decimation using Matlab.