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  • pg 1
 Multirate Signal Processing
Sampling Rate Conversion: Decimation
              S. R. M. Prasanna

                Dept of ECE,
                IIT Guwahati,

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

                                                  Multirate Signal Processing – p. 3/1
           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
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
            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
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
Thus y(m) = v(mM ) =       k=−∞ h(k)x(mM    − k)

                                                   Multirate Signal Processing – p. 7/1
      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

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
Frequency domain chrs. (contd.(1))
  ∞                    M −1 j2πmk −m/M
  m=−∞ v(m)(1/M        k=0 e     )z

       (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
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
 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
  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
Cancelation of Aliasing Effect (contd.)
    Add all these shifted stretched versions to the
    unshifted stretched version X(ejwy /M ), and divide by
    The stretched version X(ejwy /M ) can in general
    overlap with its shifted replicas. This leads to
    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
MSP-Lab1: M-fold Decimation
  To study the effect of aliasing in sampling rate
  To study the effect of frequency response of different
  order Low Pass (LP) filters.
   Consider an audio signal of CD quality (i.e., sampled
   at 44.1 kHz).
   Resample using a suitable matlab function at 16
   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
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
     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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