Multirate signal processing and decimation interpolation

1. Electronics
National
Digital
UNDER THE GUIDANCE OF
DR. RAM KUMAR KARSH SIR,
DEPT OF ELECTRONICS AND COMMUNICATION
NIT SILCHAR
1
Department of
Electronics and Communication Engineering
ational Institute of Technology
2020
Digital Signal Processing
Mini Project
Submitted By:
Group 16
RanSher (1814110)
COMMUNICATION ENGINEERING,
Engineering
Technology, Silchar

2. 2
Introduction to Multirate signal processing
and Decimation Interpolation
Ran Sher
B. Tech, 5th Semester, ECE-B, NIT Silchar
ABSTRACT
Multirate Signal Processing systems are used to improve the performance or to
increase the computational efficiency. The approach through this report is to
theoretical study of the Multirate signal processing with the two popular operations
decimation and interpolation and to identify the various aspects of multirate signal
processing and with its importance and applications in digital signal processing.
TABLE OF CONTENTS
Introduction……………………………………2
Multirate signal Processing……………………3
Decimation………………………………….…3
Interpolation…..……………………….............5
Applications of Multirate DSP……..………….8
Conclusions……………………………………8
Acknowledgement……………………………..8
References……………………………………...9
1. INTRODUCTION
A digital signal processing system that uses signals with different sampling
frequencies is probably performing multirate digital signal processing. Multirate
signal processing often uses sample rate conversion to convert from one sampling
frequency to another sampling frequency. The two basic operations in a multirate
signal processing used are decreasing sampling-rate of a signal called decimation
and increasing sampling-rate of a signal called interpolation. Multirate signal
processing systems are sometimes used for sampling-rate conversion, which
involves both decimation and interpolation.
Changing the sampling frequency in the analog domain requires digital to analog
conversion and then analog to digital conversion at a different sampling frequency.

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Both digital to analog conversion and Analog to digital conversion introduce errors
and noise into the signal. Therefore sample rate conversion is done in digital
domain with the help of methods Decimation and Interpolation.
2. MULTIRATE SIGNAL PROCESSING
In multirate digital signal processing, the sampling rate of a signal is changed in
order to increase the efficiency of various signal processing operations. Decimation,
or down-sampling, reduces the sampling rate, whereas expansion, or up-sampling,
followed by interpolation increases the sampling rate
A digital signal processing system that uses signals with different sampling
frequencies is probably performing multirate digital signal processing. Multirate
digital signal processing often uses sample rate conversion to convert from One
sampling frequency to another sampling frequency.
Sample rate conversion uses
 Decimation to decrease the sampling rate,
 Interpolation to increase the sampling rate.
Sample Rate Conversion
Changing the sampling frequency in the analog domain requires: Digital to analog
conversion then Analog to digital conversion at a different sampling frequency.
Both Digital to analog conversion and Analog to digital conversion introduce errors
and noise into the signal.
Therefore sample rate conversion is done in digital domain and uses a combination
of:
 Decimation,
 Interpolation.
3. DECIMATION
Decimation removes samples from a signal.
Decimation can therefore only down sample the signal by an integer factor:
So that,
Where D is an integer,
is the old sampling rate (number of samples/second) and is the new
sampling rate.

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Pictorial Representation of decimation
Properties of Decimation
 Decimation decreases the sampling rate.
 The sampling theorem states that the highest frequency in a signal should
be less than half the sampling frequency.
 A digital anti-aliasing filter has to be applied to remove frequencies
higher than:
 So in digital frequency, the cut-off frequency is:
Anti-aliasing Filter for Decimation
There is the need of signals to be passed to anti-aliasing filter to remove the
problem of aliasing. This means that the signal has to be filtered in the digital
domain before decimation:

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4. INTERPOLTION
Interpolation is the process of estimating unknown values that fall between known
values.
In this example, a straight line passes through two points of known value. You can
estimate the point of unknown value because it appears to be midway between the
other two points. The interpolated value of the middle point could be 9.5.
Pictorial Representation of interepolation
The new sampling frequency is greater than the old sampling frequency:
Where is the old sampling frequency and is the new sampling frequency.
Also, the new sampling frequency has to be an integer multiple of the original
sampling frequency:

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Where D is an integer.
A common interpolation approach is zero filling based interpolation.
There are two stages:
1. Zero filling: Zero filling interpolation (ZIP) is the substitution of zeroes for
unmeasured data points in order to increase the matrix size of the new data
prior to Fourier transformation of MR data.
2. Low pass filtering: A low-pass filter (LPF) is a filter that passes signals with
a frequency lower than a selected cutoff frequency and attenuates signals with
frequencies higher than the cutoff frequency.
Example 1: Interpolating by × 3 (two zero samples are inserted between each
original sample).
Non-Integer Sample Rate Conversion
Both:
 Decimation (for down sampling):
 And Interpolation (for up sampling):
Where D is an integer, can only change the sampling frequency to an integer
of the original frequency.

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Example 2:
 A CD player stores music at 44.1 kHz.
 A professional music recording device processes audio at 48 kHz.
 Transfer of the music to or from the CD player and the professional audio
device using:
 decimation only or
 interpolation only
 Are not possible because:
which is not an integer.
Combine decimation and interpolation to get non-integer sample rate conversion.
The sample rate conversion is then:
Example 3: Get audio from 44.1 kHz sampled source (CD player) and transfer to
professional audio processor requiring 48 kHz sample rate.
This process requires up sampling to 48 kHz from 44.1 kHz
1. Worst case common factor: L = 48 kHz to give
fs × 48 kHz = 2116.8 MHz.
Better alternative is L = 160 to give
L × 44.1 kHz = 7056 kHz
2. So interpolate by factor L by inserting 159 zeros for each sample in 44.1 kHz
CD player signal then low pass filtering.
3. Then decimate to 48 kHz by removing 146 samples in every 147 (= L × 44.1
kHz/48kHz) from the up sampled signal
(After applying anti-aliasing low pass filter).
The resulting sample rate conversion is:

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5. APPLICATIONS OF MULTIRATE SIGNAL PROCESSING
Some applications of multirate signal processing are
 Up-sampling
 Various systems in digital audio signal processing often operate at
different sam-pling rates. The connection of such systems requires a
conversion of sampling rate.
 Speech processing
 In the implementation of high-performance filtering operations, where a
very narrow transition band is required.
 Filer banks and wavelet transforms depends on multirate method.
 A/D and D/A converters
 Interpolation
 Used to change the rate of signal.
6. CONCLUSIONS
After studying the multirate signal processing , decimation and interpolation,
we came to the conclusion that there is the need of changing the sampling
frequency during processing of signal in order increase the efficiency which
can be possible through decimation and interpolation.
7. ACKNOWLEDGEMENT
The authors are thankful to Dr. Ram Kumar Karsh Sir, Assistant Professor,
National Institute of Technology Silchar for his guide, advise and motivation
throughout the completion of this report.
Authors would also like to express their gratitude to their college, National
Institute of Technology, Silchar for providing with such a strong platform and
enabling to harness their talents. Lastly, authors would like to express their
appreciation to their parents for providing them moral support and
encouragement.

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8. REFERENCES
[1] Digital signal processing principles, algorithms and Applications.pdf
[2] A.V.Oppenheim, R.W. Schafer and J.R. Buck, Discrete-Time Signal
Processing.pdf
[3] M. H. Hayes, Digital Signal Processing, Schaums out lines.pdf
[4] https://www.slideshare.net
[6] https://www.electronicshub.org
[7] https://en.wikibooks.org/
[8] https://en.wikipedia.org