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8/8/2019 Slides i Intro Dsp
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Fundamentals of digital signalprocessing
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Sound modeling
sound
symbols
aims
analysis
synthesis
processing
classification:
signal models
source models
abstract models
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0 500-0.05
0
0.05
x(t)
t in s ec
0 10 20-0.05
0
0.05
n
x(n)
0 10 20-0.05
0
0.05
n
y(n)
0 500-0.05
0
0.05
t in s ec
y(t)
Digital signals
analog
sampling
processingreconstruction
sampling interval T
sampling frequency fs= 1/T
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Digital signals: time representations
8000 samples
100 samples
line with dots
vertical quantization
integere.g. -32768 .. 32767
normalizede.g. -1 .. (1-Q)Q = quantization step
0 1000 2000 3000 4000 5000 6000 7000 8000-0.5
0
0.5
x(n)
0 100 200 300 400 500 600 700 800 900 1000
-0.5
0
0.5
x(n)
0 10 20 30 40 50 60 70 80 90 100-0.05
0
0.05
x(n)
n
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Spectrum: analog vs. digital signal
sampling leads to a replication of the baseband spectrum
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Spectrum: analog vs. digital signal
Sampling leads to a replication of the analog signal spectrum
Reconstruction of the analog signal:
low pass filtering the digital signal
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Discrete Fourier Transform
Magnitude
Phase
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0 0.5 1 1.5 2 2.5 3 3.5
x 104
-40
-20
0
20
|X(f)|indB
f in Hz
Magnitude s pectrum |X(f)| in dB
Discrete Fourier Transform (example)
FFT with 16 points
cosine (16 points)
magnitude (16 points)
normalization:0 dB for sinusoid 1
magnitude(frequency points)
step fs/N
magnitude dB vs. Hz
0 2 4 6 8 10 12 14 16
-1
0
1
a)
n
Cos ine s ignal x(n)
0 2 4 6 8 10 12 14 16
0
0.5
1
b)
k
Magnitude s pectrum |X(k)|
0 0.5 1 1.5 2 2.5 3 3.5
x 104
0
0.5
1
c)
f in Hz
Magnitude s pectrum |X(f)|
Nkfs/
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Inverse Discrete Fourier Transform (DFT)
if X(k) = X*(N-k)then IDFT gives N discrete-time real values x(n)
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Frequency resolution
Zero padding: to increasefrequency resolution
0 2 4 6
-1
0
1
2
x(n)
8 sa mples
0 2 4 6
0
2
4
6
8
10
|X(k)|
8-point FFT
0 5 10 15
-1
0
1
2
n
x(n)
8 samples + zero-padding
0 5 10 15
0
2
4
6
8
10
k
|X(k)|
16-point FFT
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Window functions
to reduce leakage:
weight audio samples bya window
Hamming window
wH(n) = 0.54 0.46 cos(2n/N)
Blackman window
wB(n) = 0.42 0.5cos(2n/N) + 0.08(4n/N)
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Window
Reduction of the leakage effect bywindow functions:
(a) the original signal,(b) the Blackman window function
of length N=8,
(c) product x(n)w(n) with 0n N-i,(d) zero-padding applied to z(n)
w(n) up to length N= 16
The corresponding spectra areshown on the right side.
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Spectrograms
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Waterfall representation
0 1000 2000 3000 4000 5000 6000 7000 8000-1
-0.5
0
0.5
1
Signal x(n)
x(n)
n
0 5 10 15 20
0
2000
4000
6000
-100
-50
0
Waterfall Representation of Short-time FFTs
f in Hz
n
Magnitudein
dB
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Digital systems
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Definitions
Unit impulse
Impulse reponse h(n) = output to a unit impulseh(n) describes the digital sistem
Discrete convolution:y(n)=x(n)*h(n)
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Algorithms and signal graphs
Delay
e.g. y(n) = x(n-2)
Weighting factor
e.g. y(n) = a x(n)
Addition
e.g.y(n) = a1 x(n) + a2 x(n)
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Simple digital system
weighted sum over several input samples
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Transforms
Frequency domain desciption of the digital system
Z transform
Discrete time Fouriertransform
Transfer function H(z):Z transform of h(n)
Frequency response:Discrete time Fouriertransform of h(n)
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Causal and stable systems
Causality: a discrete-time system is causal
if the output signal y(n) = 0 for n
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IIR systems
= system with infinte impulse response
e.g. second order IIR system
Difference equation
Transfer function
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IIR systems
= system with infinte impulse response h(n)
Difference equation
Z transform of diff. eq.
Transfer function
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FIR systems
= system with finite impulse response h(n)
e.g. second order FIR system
Difference equation
Z transform of diff. eq.
Transfer function
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Fir example
computation of frequency response
impulse response
magnitude resp.
pole/zero plot
phase resp. 0 2 4-0.1
0
0.1
0.2
0.3
(a) Impuls e Re s ponse h(n)
n
-1 0 1 2
-1
0
1
Re(z)
Im(z)
(c) Pole/Zero plot
4
0 10 20 30 400
0.2
0.4
0.6
0.8
f in kHz
|H(f)|
(b) Magnitude Res pons e |H(f)|
0 10 20 30 40
-2
-1.5
-1
-0.5
0
f in kHz
H(f)/
(d) Phase Response H(f)
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