ELEC3305 Digital Signal Processing
Edmund Tse 2008
s a ar ar 2 . ar n1
) rs
ar ar 2 . ar n1 ar n
o Calculated using geometric series:
1 r s a ar n
Properties of Z transform
Linearity
Right shift
x[n q+ z-qX(z) + (z-q+1x[-1] + z-q+2x[-2] + . + x[-q]
o For
ELEC3305 Digital Signal Processing
Edmund Tse 2008
xt
c
k
Fourier series
ck
1
T0
T0
k
xn
e jk0t
xt e jk0t dt
xnc
k
e jk 0 n
N
c k xne jk 0 n
DFS
N
2
0
T0
Laplace transform
1
N
0
2
N
X s xt e st dt
Z-transform
0
xz xnz n
n 0
DFS decomposes digital s
ELEC3305 Digital Signal Processing
Edmund Tse 2008
Reduce quantisation error by making errors lie symmetrically around the diagonal
o Mean squared value of error is Q2/12 centred at zero
Can easily be extended to bipolar values.
o If we offset the values
ELEC3305 Digital Signal Processing
Edmund Tse 2008
Stability of continuous signals:
Stability of discrete signals:
Chapter 4 Difference Equations
4.1 Filtering basics
Filters are systems that change a signal by altering its frequency characteristic
Descri
ELEC3305 Digital Signal Processing
Edmund Tse 2008
Digital Signal Processing
General
Lecturer
o Dr. Yash Shrivastava
o EE 305
o [email protected]
Assessments
o 30%
Practical work
o 70%
Written exam
Reference Book
o van de Vegte - Fundamentals of Digital
ELEC3305 Digital Signal Processing
Edmund Tse 2008
4.7 Impulse response
h[n] response of the filter to an impulse input
Provides a complete description of the filter
Infinite impulse response (IIR) takes infinitely long to decay to zero, typical in recurs
ELEC3305 Digital Signal Processing
Edmund Tse 2008
Illustration of impulse train sampling
o
This implicitly assumes that the sampling frequency must be at least twice the sampled bandwidth:
Nyquist Rate: S 2 B
Reconstruction of signal from sampled signal
ELEC3305 Digital Signal Processing
Edmund Tse 2008
Ideal low pass filter:
Ideal high pass filter:
Ideal band pass filter:
Ideal band stop (notch) filter:
Chapter summary
An analog signal is defined at every point in time and may take any amplitude. A digi
ELEC3305 Digital Signal Processing
Edmund Tse 2008
Notation
9.5 Windows
Rectangular
o Fourier transform of rectangular window is a sinc function periodic convolution
o Side lobe gives ~10% of passband gain may be too large for some applications
o 21dB
Han
ELEC3305 Digital Signal Processing
Edmund Tse 2008
Properties of DTFT
Linearity:
Time shift:
Time reversal:
ax[n] + bv[n+ aX() + bV()
x[n q+ X()e-jq for any integer q
x[-n+ X(-)
Multiplication by n:
nxn j
Multiplication by complex exponential:
Convolution
Q3
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