Unit-V DSP APPLICATIONS

UNIT V -SYLLABUS DSP APPLICATIONS Multirate signal processing: Decimation Interpolation Sampling rate conversion by a rational factor Adaptive Filters: Introduction Applications of adaptive filtering to equalization.

Multirate Digital Signal Processing Basic Sampling Rate Alteration Devices Up-samplerUp-sampler - Used to increase the sampling rate by an integer factor Down-samplerDown-sampler - Used to decrease the sampling rate by an integer factor

DECIMATION

Down-Sampler Time-Domain Characterization down-sampling factorAn down-sampler with a down-sampling factor M, where M is a positive integer, develops an output sequence y[n] with a sampling rate that is (1/M)-th of that of the input sequence x[n] Block-diagram representation Mx[n]x[n] y[n]y[n]

Down-Sampler Down-sampling operation is implemented by keeping every M-th sample of x[n] and removing in- between samples to generate y[n] Input-output relation y[n] = x[nM]

Down-Sampler Figure below shows the down-sampling by a factor of 3 of a sinusoidal sequence of frequency Hz obtained using Program 10_2

In downsampling by an integer factor D>1, every D-th samples of the input sequence are kept and others are removed: D Decimation by a factor D

Relationship in time domain Input sequence Periodic train of impulses Output sequence Decimation by a factor D

C Relationship in frequency domain Decimation by a factor D

C

h(n) D Using a digital low-pass filter to prevent aliasing Decimation by a factor D

INTERPOLATION

Up-Sampler Time-Domain Characterization up-sampling factorAn up-sampler with an up-sampling factor L, where L is a positive integer, develops an output sequence with a sampling rate that is L times larger than that of the input sequence x[n] Block-diagram representation Lx[n]x[n]

Up-Sampler Up-sampling operation is implemented by inserting equidistant zero- valued samples between two consecutive samples of x[n] Input-output relation

Up-Sampler Figure below shows the up-sampling by a factor of 3 of a sinusoidal sequence with a frequency of 0.12 Hz obtained using Program 10_1

Up-Sampler In practice, the zero-valued samples inserted by the up-sampler are replaced with appropriate nonzero values using some type of filtering process interpolationProcess is called interpolation and will be discussed later

In up-sampling by an integer factor I >1, I -1 equidistant zeros-valued samples are inserted between each two consecutive samples of the input sequence. Then a digital low-pass filter is applied. I h(n) Interpolation by a factor I

Relationship in frequency domain Input sequence Interpolation by a factor I

If is a rational number h 1 (n ) h 2 (n ) I D interpolationdecimation Sampling rate conversion by a rational factor I/D Sampling period

h (n) I D Sampling Rate Conversion

INTRODUCTION TO ADAPTIVE FILTER

Adaptive filter the signal and/or noise characteristics are often nonstationary and the statistical parameters vary with time An adaptive filter has an adaptation algorithm, that is meant to monitor the environment and vary the filter transfer function accordingly based in the actual signals received, attempts to find the optimum filter design

ADAPTIVE FILTER The basic operation now involves two processes : 1. a filtering process, which produces an output signal in response to a given input signal. 2. an adaptation process, which aims to adjust the filter parameters (filter transfer function) to the (possibly time-varying) environment Often, the (avarage) square value of the error signal is used as the optimization criterion

Adaptive filter Because of complexity of the optimizing algorithms most adaptive filters are digital filters that perform digital signal processing When processing analog signals, the adaptive filter is then preceded by A/D and D/A convertors.

Adaptive filters differ from other filters such as FIR and IIR in the sense that: –The coefficients are not determined by a set of desired specifications. –The coefficients are not fixed. With adaptive filters the specifications are not known and change with time. Applications include: process control, medical instrumentation, speech processing, echo and noise calculation and channel equalisation.

Introduction To construct an adaptive filter the following selections have to be made: –Which method to use to update the coefficients of the selected filter. –Whether to use an FIR or IIR filter.

44 Adaptive filter The generalization to adaptive IIR filters leads to stability problems It’s common to use a FIR digital filter with adjustable coefficients. [1]

45 LMS Algorithm Most popular adaptation algorithm is LMS Define cost function as mean-squared error Based on the method of steepest descent Move towards the minimum on the error surface to get to minimum gradient of the error surface estimated at every iteration [2]

46 LMS Algorithm [2]

47 Stability of LMS The LMS algorithm is convergent in the mean square if and only if the step-size parameter satisfy Here max is the largest eigenvalue of the correlation matrix of the input data More practical test for stability is Larger values for step size –Increases adaptation rate (faster adaptation) –Increases residual mean-squared error [2]

48 Applications of Adaptive Filters: Identification Used to provide a linear model of an unknown plant Applications: –System identification [2]

49 Applications of Adaptive Filters: Inverse Modeling Used to provide an inverse model of an unknown plant Applications: –Equalization (communications channels) [2]

50 Applications of Adaptive Filters: Prediction Used to provide a prediction of the present value of a random signal Applications: –Linear predictive coding [2]

51 Applications of Adaptive Filters: Interference Cancellation Used to cancel unknown interference from a primary signal Applications: –Echo / Noise cancellation hands-free carphone, aircraft headphones etc [2]

Example: Acoustic Echo Cancellation 52 [1]