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This deals about the digital processing programs

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Multirate Signal Processing
Sampling Rate Conversion: Decimation
S. R. M. Prasanna

Dept of ECE,
IIT Guwahati,
prasanna@iitg.ernet.in

Multirate Signal Processing – p. 1/1
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Need for Sampling Rate Conversion
Sampled signal at one sampling frequency needs to be
used in a system operating at different sampling
frequency. Studio (48 kHz) to CD (44.1 kHz) quality
audio.
In applications to deal with signals that are sampled at
different sampling rates. Telecommunication system
that deals with speech, audio, fascimile, video etc.
Signals may need to be processed at different sampling
rates at various stages. Signal at one rate needs to be
compared with signal at other rate.
Signal compression by subband coding.

Multirate Signal Processing – p. 2/1
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Sampling Rate Conversion
Converting signal from given rate to different rate
Analog Sampling rate conversion: DAC, analog domain
Digital sampling rate conversion: Digital to Digital
Advantage of digital conversion: less distortion and
complexity
Types of Digital Sampling Rate Conversion
M-fold Decimation
L-fold interpolation
L/M-fold fractional rate conversion

Multirate Signal Processing – p. 3/1
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M-fold Decimation
M-fold decimation implies downsampling by M
For every M samples of x(n), only one sample is
selected as output y(m)
Mathematically, y(m) = x(mM ), m = 0, 1, 2, ...
Compression in time domain by factor M leads to
expansion of the spectra by M

Multirate Signal Processing – p. 4/1
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Mapping from Linear to Angular Freq.
Depends on the sampling frequency
From sampling theorem, Fmax ≤ Fs /2, where Fs is the
sampling freq and let Fmax = Fs /2
For −Fmax ≤ F ≤ Fmax , −π ≤ ω ≤ π
Mapping Process:
2π(F/Fs ) = ω
I/P of decimator, 2π(F/Fx ) = ωx
O/P of decimator, 2π(F/Fy ) = ωy
ωy = 2π(F/Fy ) = M 2π(F/Fx ) = M ωx
ωy = M ωx

Multirate Signal Processing – p. 5/1
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Mapping (contd.)
Let, F = Fmax /M , ωy = M 2π(Fmax /M Fx )
For Fmax /M ≤ F ≤ Fmax /M , −π ≤ ωy ≤ π in the output
of the decimator
Fmax /M ≤ |F | ≤ Fmax maps outside −π to π
For 2π − Fmax /M ≤ F ≤ 2π + Fmax /M , maps from
π ≤ ω ≤ 3π in the output of the decimator
2π + Fmax /M ≤ |F | ≤ 2π + Fmax maps outside π to 3 π
Leads to aliasing of frequency components
Input signal is ﬁrst low pass ﬁltered to a cut-off freq of
Fmax /M , so that no component will spill over the interval
of observation.
After downsampling by M , freq. range of interest will be
−Fmax /M ≤ F ≤ Fmax /M
Multirate Signal Processing – p. 6/1
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Analysis of Decimation Process
An M-fold decimator will have a low pass ﬁlter followed
by an M -factor downsampler
Input sequence is x(n) with sampling frequency Fx
Output sequence is y(m) with sampling frequency Fy
Low pass ﬁlter: hd (n) ↔ Hd (ejω )
Hd (ejω ) = 1 for ω ≤ π/M and 0 otherwise
∞
Output of LPF: v(n) =     k=−∞ h(k)x(n − k)
v(n) is then downsampled by the factor M to obtain
y(m).
∞
Thus y(m) = v(mM ) =       k=−∞ h(k)x(mM    − k)

Multirate Signal Processing – p. 7/1
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Frequency domain Chrs.
Spectrum of o/p y(m) in terms of spectrum of i/p x(n)
˜
Let v (n) = v(n) for n = 0, ±M, ±2M, ... and 0 otherwise
˜
v (n) = v(n)p(n), where p(n) is a periodic train of
impulses with period M
M −1 j2πnk
By DTFS p(n) = 1/M       k=0 e
M

˜
y(m) = v (mM ) = v(mM )p(mM )
∞
z.t of output sequence Y (z) =     m=−∞ y(m)z −m
∞
Y (z) =        v (mM )z −m
m=−∞ ˜
∞
Y (z) =      v (m)z −m/M
−∞ ˜

Multirate Signal Processing – p. 8/1
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Frequency domain chrs. (contd.(1))
∞                    M −1 j2πmk −m/M
m=−∞ v(m)(1/M        k=0 e     )z
M

(M −1)   ∞         j2πmk
1/M   k=0      m=−∞ v(m)e M z −m/M
(M −1)
v(m)(e M z 1/M )−m
∞          −j2πk
1/M   k=0      m=−∞
M −1
1/M   k=0  V (e−(j2πk)/M z 1/M )
M −1
1/M   k=0  Hd (e−(j2πk)/M z 1/M )X(e−(j2πk)/M z 1/M )

Multirate Signal Processing – p. 9/1
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Frequency domain chrs. (contd.(2))
Let z = ejωy
ωy −2πk     ωy −2πk
M −1
Y (ejωy )   = 1/M   k=0 Hd (e
M    )X(e M )
with a properly designed ﬁlter Hd (ejω ), the aliasing is
eliminated and consequently all but the ﬁrst term will
vanish
Y (ejωy ) = 1/M (Hd (ωy /M )X(ωy /M ))
Y (ejωy ) = 1/M (X(ωy /M )) for 0 ≤ ωy ≤ π

Multirate Signal Processing – p. 10/1
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Cancelation of Aliasing Effect
The expression for the output of M-fold decimator is
given by
ωy −2πk     ωy −2πk
M −1
Y (ejωy )   = 1/M   k=0 Hd (e
M    )X(e M )
Let us assume that there is no low-pass ﬁlter and hence
direct decimation.
ωy −2πk
M −1
Y (ejωy )   = 1/M   k=0 X(e
M    )
This eqn can be interpreted graphically as follows
Stretch X(ejωx ) by a factor M to obtain X(ejωy /M ).
Create (M − 1) copies of this stretched version by
shifting it uniformly in successive amounts of 2π .

Multirate Signal Processing – p. 11/1
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Cancelation of Aliasing Effect (contd.)
Add all these shifted stretched versions to the
unshifted stretched version X(ejwy /M ), and divide by
M
The stretched version X(ejwy /M ) can in general
overlap with its shifted replicas. This leads to
aliasing.
Aliasing can be avoided if x[n] is a low-pass signal
band limited to the region ω < π/M .

Multirate Signal Processing – p. 12/1
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MSP-Lab1: M-fold Decimation
Objectives:
To study the effect of aliasing in sampling rate
reduction.
To study the effect of frequency response of different
order Low Pass (LP) ﬁlters.
Steps:
Consider an audio signal of CD quality (i.e., sampled
at 44.1 kHz).
Resample using a suitable matlab function at 16
kHz.
Take a 50-100 ms segment of the audio signal in one
of the high energy regions.
Compute its linear and log magnitude spectra.
Plot the time domain signal and its spectra.
Multirate Signal Processing – p. 13/1
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MSP-Lab1: M-fold Decimation (contd.)
Design an FIR LPF by windowing method for a cutoff
frequency Fc = 4kHz.
Plot its impulse response and frequency response.
Obtain LP ﬁltered signal by convolution.
Plot the time domain signal and its spectra of LPF
output,
Choose every 2nd sample to obtain decimated
signal and plot its spectra.
Choose every 2nd sample of original audio signal
(sampled at 16 kHz) to obtain decimated signal with
aliasing and plot its spectra.
Repeat sampling rate reduction process with
different orders for low pass ﬁlter.
Comment on sampling rate reduction process by
comparing results at various stages.      Multirate Signal Processing – p. 14/1

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