Discrete version of a differential equation

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Discrete version of a differential equation
Approximating the derivative (backward
difference) with discrete values at time
intervals n?
and writing as sequences
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Describing a Discrete Systemin terms of Input
and Output Signals
mn
yn
H(z)
Difference equation yn a1yn-1
akyn-k blmn b1mn-1 blmn-l
Convolution Sum
Transfer function H(z) Y(z) / X(z) Frequency
response H(ej?) Y(ej?) / X(ej?)
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A recursive system
A non-recursive system
yn0.25 xn0.5xn-10.25 xn-2 The
z-transform is The frequency response is
yn 0.9 yn-1 xn - xn-1 The
z-transform is The frequency response is

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Output of a discrete system
Consider the 3-point smoother hn ¼xn
½xn-1 ¼xn-2 with input xn a discrete
pulse xn dn dn - 1 The output yn
can be computed by convolving hn with xn
Energy 12 12 2
Energy 0.252 0.752 0.752 0.252 1.25
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The Unit Circle in the z-plane
Imag z
zej?/2 j
z-plane
zej?/4 (1j)/?2
zej0 1
zej? -1
?
Real z
zej2? 1
unit circle
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Things to know about z-Transforms
How pole and zero locations affect system
stability Transfer function of discrete
systems Frequency response (magnitude and
phase) From transfer function From
discrete-time Fourier transform Periodicity of
frequency response and relation to half the
sampling frequency
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Discrete Fourier Transformsamples the
discrete-time Fourier transform
Direct Transform
z-plane
Imag z
X
X
X
Real z
X
X
Inverse Transform
X
X
X
Illustrates N 8
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8-point Discrete Fourier Transform
xn sin(2?n/8)
Xk 0 -j4 0 0 0 0 0 j4
0 (0.5 - j0.5) (-j) ( -0.5 - j0.5) 0
(0.5 - j0.5) (-j) (-0.5 - j0.5) -j4
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8-point inverse Discrete Fourier Transform
Xk 0 -j4 0 0 0 0 0 -j4
xn 0 0.7071 1.0000 0.7071 0.0000
-0.7071 -1.0000 -0.7071
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Periodicity in discrete Fourier Series
time (seconds)
0
0.05
0.1
-0.05
2?
-2?
frequency (Hz)
0
4?
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This is what we work with!
fs 160 Hz
Upper half of sequence represents the negative
frequencies
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Sampling a continuous signal
x(t) 0.2 sin(2? 20t)
Sampling frequency fs 160 Hz Ts 1/160
xn x(t)tnTs 0.2 sin (2p 20n/160)
0.2 sin (n 2p / 8) Period N 8
and Discrete frequency 2 p/N p/4
rad Discrete frequency as a fraction of the
sampling rate is 2 p(20/160) p/4 Sampling
frequency corresponds to a discrete frequency of
2 p rad.
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Frequency axes for the DFT
16 points sampled at 800 Hz gives 20 millisec of
data
MATLAB
n 16 corresponds to ? 2?-?? which
corresponds to 800 Hz-?f
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Demonstration of aliasing caused by sampling at
too low a rate
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Zero-extended Spectrum
length(x) 32 fx fft(x)intx
zeros(1,732)fxlong fx(116) intx
fx(1732)xlong (256/32)real(ifft(fxlong))
Need to adjust amplitude so that inverted signal
has same energy as original signal.
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Power Spectra (periodic sequences) orEnergy
Spectra (finite sequences)
where INk is the periodogram and Xk are the
magnitudes (real) of the DFT coefficients