dsp processors-iii

DSP PROCESSORS-III : 

DSP PROCESSORS-III

Slide 2: 

Module 3

Syllabus : 

Syllabus Digital Filters IIR FIR Adaptive filters DIT and DIF algorithms

Discrete-time system : 

Discrete-time system Discrete-time system has discrete-time input and output signals

Slide 7: 

A discrete-time filter is a discrete-time system that passes certain frequency components and stops others. A Digital filter can be either an IIR or FIR filter. An IIR filter usually requires less cost, i.e., less computation and memory. However, an FIR filter usually has better performance, especially in the phase response( a generalized linear phase is required). Discrete-Time Filters

IIR filter : 

IIR filter IIR filter have infinite-duration impulse responses, hence they can be matched to analog filters, all of which generally have infinitely long impulse responses.

The main problem : 

The main problem There is no control over the phase characteristics of the IIR filter. Hence IIR filter designs will be treated as magnitude-only designs.

Slide 10: 

Analysis of Ideal Filters Consider an ideal low pass filter. Over period (-, ), the frequency response of an ideal low pass filter is defined as Let xi(n)=Aiexp (jin) be a frequency component of the input signal. Then, the corresponding output signal of this filter is yi(n)=Ai exp (jin) H (i). If |i|  c , then yi(n)=Aiexp [ji (n-)]. ie, xi(n) is passed with a constant delay .

Slide 11: 

If |i|>c , then yi(n)=0. ie, xi(n) is stopped. The impulse response of the ideal low pass filter is Since h(n)0 for n<0, the ideal low pass filter is non-causal and cannot be implemented in real time. However, in practical applications, a causal low pass filter is often required. (The word causal indicates that the filter output depends only on past and present inputs) Depending on a specific application, this causal low pass filter can be an IIR or FIR filter.

Prototype Analog Filters : 

Prototype Analog Filters IIR filter design techniques rely on existing analog filters to obtain digital filters. We designate these analog filters as prototype filters. The prototypes widely used in practice Butterworth lowpass Chebyshev lowpass (Type I and II)

Butterworth Lowpass Filter : 

Butterworth Lowpass Filter This filter is characterized by the property that its magnitude response is flat in both passband and stopband. Moderate phase distortion The magnitude-squared response of an N-order lowpass filter Ωc is the cutoff frequency in rad/sec.

Slide 15: 

Plot of the magnitude-squared response

Properties of Butterworth Filter : 

Properties of Butterworth Filter Magnitude Response |Ha(0)|2 =1 |Ha(jΩc) |2 =0.5, for all N |Ha(jΩ)|2 monotonically decrease for Ω Approaches to ideal filter when N→∞

Chebyshev Lowpass Filter : 

Chebyshev Lowpass Filter Chebyshev -I filters Have equiripple response in the passband Chebyshev -II filters Have equiripple response in stopband Butterworth filters Have monotonic response in both bands We note that by choosing a filter that has an equripple rather than a monotonic behavior, we can obtain a low-order filter. (The maximum delay, in samples, used in creating each output sample is called the order of the filter ) Therefore Chebyshev filters provide lower order than Buttworth filters for the same specifications.

Slide 19: 

Chebyshev -I filters Chebyshev -II filters

The magnitude-squared response of Chebyshev -I filter : 

The magnitude-squared response of Chebyshev -I filter N is the order of the filter, Epsilon is the passband ripple factor For 0<x<1, TN(x) oscillates between –1 and 1, and (b) For 1<x<infinity, TN(x) increases monotonically to infinity

Observations : 

Observations At x=0 (or Ω=0); |Ha(j0)|2 = 1; for N odd; = 1/√(1+Є2); for N even At x=1 (or Ω= Ωc); |Ha(j1)|2 = 1/√(1+Є2); for all N. For 0<=x<=1 (or 0<= Ω<= Ωc) |Ha(jx)|2 oscillates between 1 and 1 /√(1+Є2) For x>1 (or Ω > Ωc), |Ha(jx)|2 decreases monotonically to 0 .

Analog-to-digital Filter Transformations : 

Analog-to-digital Filter Transformations After designing various analog filters, now transform them into digital filters. These transformations are derived by preserving different aspects of analog and digital filters. Impulse invariance transformation Preserve the shape of the impulse response from Analog to Digital filter Finite difference approximation technique Convert a differential eq. representation into a corresponding difference eq. Step invariance Preserve the shape of the step response Bilinear transformation Preserve the system function representation from Analog to Digital domain

Impulse Invariance Transformation : 

Impulse Invariance Transformation The digital filter impulse response look similar to that of a frequency-selective analog filter. basic principle is sampling of impulse response of an analog filter Sample ha(t) at some sampling interval T to obtain h(n) h(n)=ha(nT) Since z = e jw on the unit circle and s = jΩ on the imaginary axis, we have the following transformation from the s-plane to the z-plane: z=e sT also, and (Frequency-domain aliasing formula)

Slide 27: 

Complex-plane mapping in impulse invariance transformation

Properties: : 

Properties: Sigma < 0, maps into |z| <1 (inside of the Unit Circle) Sigma = 0, maps into |z| =1 (on the Unit Circle) Sigma >0, maps into |Z| >1 (outside of the Unit circle) Causality and Stability are the same without changing Aliasing occur if filter not exactly band-limited

Let digital lowpass filter specifications be wp,ws,Rp and As. To determine H(z) , first design an equivalent analog filter and then mapping it into the desired digital filter. Design Procedure: : 

Let digital lowpass filter specifications be wp,ws,Rp and As. To determine H(z) , first design an equivalent analog filter and then mapping it into the desired digital filter. Design Procedure: 1. Choose T and determine the analog frequencies: Ωp =wp/T, Ωs = ws/T 2. Design an analog filter Ha(s) using the specifications with one of the prototypes of the previous section. 3. Using partial fraction expansion, expand Ha(s) into 4. Now transform analog poles {pk} into digital poles {epkT} to obtain the digital filter

Advantages : 

Advantages It is a stable design and the frequencies Ω and w are linearly related. Disadvantage We should expect some aliasing of the analog frequency response, and in some cases this aliasing is intolerable. This design method is useful only when the analog filter is essentially band-limited to a lowpass or band pass filter in which there are no oscillations in the stopband.

Bilinear Transformation : 

Bilinear Transformation transforms continuous-time system representations to discrete-time and vice versa This mapping is the best transformation method. It maps positions on the jω axis, in the s-plane to the unit circle in the z-plane For every feature in the frequency response of the analog filter, there is a corresponding feature, with identical gain and phase shift, in the frequency response of the digital filter basic principle: application of the trapezoidal formula for numerical integration of differential equation

Complex-plane mapping in bilinear transformation : 

Complex-plane mapping in bilinear transformation

Observations : 

Observations Sigma < 0 |z| < 1, Sigma = 0 |z| = 1, Sigma > 0 |z| > 1 The entire left half-plane maps into the inside of the Unit circle. This is a stable transformation. The imaginary axis maps onto the Unit circle in a one-to-one fashion. Hence there is no aliasing in the frequency domain. Relation of ω to Ω is nonlinear ω = 2tan-1(ΩT/2) ↔ Ω = 2tan(ω/2)/T;

Given the digital filter specifications wp,ws,Rp and As, we want to determine H(z). The design steps: : 

Given the digital filter specifications wp,ws,Rp and As, we want to determine H(z). The design steps: Choose a value for T. this is arbitrary, and we may set T = 1. Prewarp the cutoff freq. wp and ws; ie, calculate Ωp and Ωs using Ωp = 2/T*tan(wp/2), Ωs = 2/T*tan(ws/2) 3. Design an analog filter Ha(s) to meet the specifications. 4. Finally set And simplify to obtain H(z) as a rational function in z-1

Advantage of the bilinear Transformation : 

Advantage of the bilinear Transformation It is a stable design There is no aliasing There is no restriction on the type of filter that can be transformed.

FIR FILTER : 

FIR FILTER A finite impulse response filter is a type of a digital filter. The impulse response is 'finite' because it settles to zero in a finite number of sample intervals. The impulse response of an Nth order FIR filter lasts for N+1 samples, and then dies to zero. Properties: Inherently stable - all the poles are located at the origin and thus are located within the unit circle. Require no feedback- any rounding errors are not compounded by summed iterations. This also makes implementation simpler. can be designed to be linear phase - which means the phase change is proportional to the frequency.

Slide 37: 

Disadvantage: considerably more computation power is required compared with a similar IIR filter. This is especially true when low frequencies are to be affected by the filter. Design methods: Window design method Frequency Sampling method Weighted least squares design Minimax design Equiripple design

Window design method : 

Window design method Simplest FIR filter design is window function technique Specific window should be chosen according to desired constraints the impulse response of the causal FIR filter is obtained by windowing the ideal filter.

Slide 39: 

windowing hd(n), we can obtain the impulse response of a causal FIR lowpass filter. ie, h(n) = hd(n) w(n) where w(n) is a window function and is equal to 0 for n<0. Generally, w(n) is real and symmetric, and has a maximum value of unity.  is chosen as ,  = (N-1)/2, where N is the length of w(n).

Slide 40: 

n hd(n) w(n) n h(n) n Frequency Response

Slide 41: 

Window functions The rectangular window is defined as The Bartlett window is defined as

Slide 42: 

The Hamming window is defined as The Blackman window is defined as The Hann window is defined as

Slide 43: 

Bartlett Window Rectangular Window Hann Window

Slide 44: 

Blackman Window Hamming Window

Slide 45: 

It can be seen that when the length of the window is fixed, a narrow transition band corresponds to large ripples Features of Commonly Used Windows.

Slide 46: 

The Kaiser window is defined as Here, I0(·) is the zero-order modified Bessel function of the first kind and  is a shape parameter.  should be selected to obtain a good tradeoff between the width of the transition band and the amplitudes of the ripples.

Slide 47: 

The windowing method is carried out in the following steps: 1. Determine the type of the window according to the specification for the amplitudes of the ripples. 2. Find the length of the window according to the specification for the width of the transition band. 3. Find  4. Determine the impulse response of the causal FIR lowpass filter

Adaptive filter : 

Adaptive filter An adaptive filter is a filter that self-adjusts its transfer function according to an optimizing algorithm. Because of the complexity of the optimizing algorithms, most adaptive filters are digital filters that perform digital signal processing and adapt their performance based on the input signal. In IIR and FIR filters process parameters are known in advance or variation is assumed to be known Adaptive filters are best suited when signal conditions are slowly changing or there is large uncertainty and filter has to compensate for that

Slide 49: 

Here the adaptive filter output y is compared with a desired signal d to yield a error signal e which is fed back to adaptive filter Coefficients of adaptive filter is adjusted or optimized using a least mean square algorithm based on the error signal Adaptive filter d x + - e y Basic adaptive filter:

Slide 50: 

Output of adaptive filter: y(n)=Σ wk(n) x(n-k) where wk(n) represent N weights for a specific time n. The performance measure is based on the error signal. The error signal, e(n) = d(n)-y(n) The weights of wk(n) are adjusted such that a mean squared error function is minimized Mean square error function is E[e2(n)],where E represents the expected value K=0 N-1

Least mean square algorithm: : 

Least mean square algorithm: The simple and effective means of updating the weights without averaging or differentiating Wk(n+1) = wk(n)+2βe(n). x(n-k) , k=0,1,…,N-1 Where x(n) is the input and β is adaptive step size. For each time n, weight wk(n) is updated or replaced by new coefficients, unless the error function is zero. After the output y(n), error signal e(n) and wk(n) are updated for a specific time n, new sample is taken and the adaptation process is repeated

Slide 52: 

Applications of adaptive filters Noise cancellation Signal prediction Adaptive feedback cancellation Echo cancellation

For noise cancellation: : 

For noise cancellation: The basic adaptive structure is modified for a noise cancellation system Adaptive filter d+n n’ + - e The desired signal d is corrupted by noise n. The input to the adaptive filter is noise n’ that is correlated with noise n (noise n’ could come from same source as n but modified by environment) The output y is adapted to noise n When this happens the error signal approaches to desired signal d. The overall output is e and not the adaptive filter output y. y

For system identification: : 

For system identification: The same input u is given to the unknown system (plant) and to the adaptive filter. The error signal e is the difference between response d and the filter output y. The error signal e is fed to the adaptive filter and is used to update the filter coefficients, until y = d When this happens the adaption process is finished and e approaches to zero. Thus the unknown system is modeled.

Signal Prediction: : 

Signal Prediction: Used to provide a prediction of the present value of a random signal This is like interference cancellation, but the adaptive filter uses a delayed version of the primary signal as the reference. Parameters u = input of adaptive filter = delayed version of random signal y = output of adaptive filter d = desired response = random signal e = d - y =estimation error = system output

DFT : 

DFT The discrete Fourier transform (DFT) is one of the specific forms of Fourier analysis. It transforms one function into another, which is called the frequency domain representation of the original function (the time domain). IDFT does the inverse operation But the DFT requires an input function that is discrete and whose non-zero values have a limited (finite) duration. Such inputs are often created by sampling a continuous function, like a person's voice

Slide 58: 

The FFT is based on the divide-and-conquer paradigm

The DIF FFT is the transpose of the DIT FFT : 

The DIF FFT is the transpose of the DIT FFT To obtain flowgraph transposes: Reverse direction of flowgraph arrows Interchange input(s) and output(s) DIT butterfly: DIF butterfly:

Slide 72: 

Comparing DIT and DIF structures: DIT FFT structure: DIF FFT structure:

Slide 73: 

Steps involved in DIF FFT algorithm: Data shuffling not required N=2p & input is separated into two parts . The first set consists of first N/2 input samples with n ranging from 0 to N/2-1 Second set consists of the remaining N/2 input samples with n ranging from N/2 to N-1 X(k) is decimated into even & odd numbered points (DIF) decomposing an N point DFT into 2 N/2Point DFTs The procedure is repeated till we get a 2-point DFT. The 2-point DFT is then converted to butterfly to get the complete structure.

DIF or DIT? : 

DIF or DIT? In terms of computational work load, both perform exactly the same number of butterflies. Each butterfly requires exactly one complex multiply and two complex adds. The most significant difference between simple DIF and DIT algorithms is that DIF starts with normal order input and generates bit reversed order output. In contrast, DIT starts with bit reversed order input and generates normal order output. So use DIF for the forward transform and DIT for the inverse transform

Slide 78: 

Advantages: (i) Reasonably good saving in terms of computation, i.e. it is efficient (ii) Short Program required (iii) Easy to understand Disadvantages: (i) A relatively large number of operations still required, especially multiplications (ii) Problem of the generation of WN for n=0,1,2,…N-1.

Applications of FFT : 

Applications of FFT Linear convolution (1) Append zeros to the two sequences of lengths N and M, to make them of lengths an integer power of two that is larger than or equal to M+N-1. (2) Apply FFT to both zero appended sequences (3) Multiply the two transformed domain sequences (4) Apply inverse FFT to the new multiplied sequence

THE END : 

THE END