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Practical Signal Processing Concepts and Algorithms using MATLAB. LTI Systems. Section Outline. LTI system representations The z-Transform Frequency and impulse response Introduction to filtering. Linear Time-Independent Systems. x(n). y(n). δ (n). h(n).

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slide2

Section Outline

  • LTI system representations
  • The z-Transform
  • Frequency and impulse response
  • Introduction to filtering
slide4

Linear System Representations:Difference Equations

>> b = [b0 b1 .. bM]; a = [a0 .. aN];

Example:

>> b = [1 2 0]; a = [1 0 0.2];

slide6

The z-Transform

x(n)  X(z)

  • Linearity and superposition are preserved
  • x(n–k) z–kX(z)
  • x(–n) X(1/z)
  • anx(n) X(z/a)
  • x(n)*y(n) X(z)Y(z)

y(n) = x(n)*h(n) Y(z) = X(z) H(z)

Transfer

function

slide7

Linear System Representations:Transfer Functions

z-transform of difference equation

Discrete filter form

Proper form

Example:

>> b = [1 2 0];

>> a = [1 0 0.2];

slide8

Filters and Transfer Functions

In general, the z-transform Y(z) of a digital filter's output y(n) is related to the z-transform X(z) of the input by

where H(z) is the filter's transfer function. Here, the constants b(i) and a(i) are the filter coefficients and the order of the filter is the maximum of n and m.

H(z)

slide9

For example let;

>>step = ones(50); %input data : step function

>>b = 1; % Numerator

>>a = [1 -0.9]; % Denominator

where the vectors b and a represent the coefficients of a filter in transfer function form. To apply this filter to your data, use

>> y = filter(b,a,step);

>> stem(y)

>>fvtool(b,a) %GUI.Don’t have to define input (if input is %step/impulse function)%

slide10

Linear System Representations:Zero-Pole-Gain

Factored proper form

Zeros:z2 + 2z = 0

z(z + 2) = 0

z = 0andz = –2

Poles:z2 + 0.2 = 0

z = ± j

Gain:k = 1

Example:

slide11

Zero-Pole-Gain Analysis

>> b = [1 2 0];

>> a = [1 0 0.2];

>> [z,p,k] = tf2zpk(b,a);

>> zplane(z,p)

slide12

Zero-Pole Placement in Filter Design

x

o

passband

stopband

stable: inside

unit circle

minimum phase:

inside unit circle

x

o

x

x

o

no effect on

magnitude response

conjugate pairs

x

o

Steep transitions: stopband os paired with xs along same radial line

Causal system: number of os ≤ number of xs

slide13

Zero-Pole-Gain Editing in SPTool

  • Choose Pole/Zero Editor design algorithm
  • View system characteristics as you edit
slide15

Impulse Response

δ(n)

h(n)

>> impz(b,a,n,fs)

frequency response
Frequency Response

>> freqz(b,a,1,fs)

frequency response and the transfer function
Frequency Responseand the Transfer Function

Frequency response

Unwrapped response

x

o

x

filter visualization tool
Filter Visualization Tool
  • Magnitude response
  • Phase response
  • Group delay
  • Impulse response
  • Step response
  • Pole-zero plot
  • Filter coefficients

>> a = [1 0 .2];

>> b = [1 2 0];

>> fvtool(b,a)

filtering a signal
Filtering a Signal

>> y = filter(b,a,x)

slide20

Filter Implementation and AnalysisConvolution and Filtering

The mathematical foundation of filtering is convolution. The MATLAB conv function performs standard one-dimensional convolution, convolving one vector with another

A digital filter's output y(k) is related to its input x(k) by convolution with its impulse response h(k).

>> x = [1 2 1];

>> h = [1 1 1];

>> y = conv(h,x);

>> stem(y)

slide21

Importing a Filter into SPTool

Choose FileImport from the SPTool main menu.

applying a filter in sptool

Applying a Filter in SPTool

Applying a Filter in SPTool

Select the signal and the filter

Choose the name of the output signal

Click on Apply

slide23

Filter Coefficients and Filter Names

In general, the z-transform Y(z) of a digital filter's output y(n) is related to the z-transform X(z) of the input by

  • Many standard names for filters reflect the number of a and b coefficients present:
    • When n=0 (that is, b is a scalar), the filter is an Infinite Impulse Response (IIR), all-pole, recursive, or autoregressive (AR) filter.
    • When m=0 (that is, a is a scalar), the filter is a Finite Impulse Response (FIR), all-zero, nonrecursive, or moving-average (MA) filter.
    • If both n and m are greater than zero, the filter is an IIR, pole-zero, recursive, or autoregressive moving-average (ARMA) filter.
slide24

IIR Filter Design

  • The primary advantage of IIR filters over FIR filters is that they typically meet a given set of specifications with a much lower filter order than a corresponding FIR filter.
slide25

Section Summary

  • LTI system representations
  • The z-transform
  • Frequency and impulse response
  • Introduction to filtering