VIEWS: 12 PAGES: 14 POSTED ON: 8/22/2011
This deals about the digital processing programs
www.jntuworld.com 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 www.jntuworld.com 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 www.jntuworld.com 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 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 www.jntuworld.com 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 www.jntuworld.com 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 www.jntuworld.com 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 www.jntuworld.com 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 www.jntuworld.com 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 www.jntuworld.com 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 www.jntuworld.com 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 www.jntuworld.com 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 www.jntuworld.com 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 www.jntuworld.com 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 www.jntuworld.com 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