Fundamentals of digital signal
processing
1
Sound modeling
sound
symbols
aims
analysis
synthesis
processing
classification:
signal models
source models
abstract models
2
Digital signals
analog
sampling
processing
reconstruction
x(t)
x(n)
y(n)
y(t)
0.05
0.05
0.05
0.05
0
0
0
0
-0.05
-0.05
500 0
0
t in µs ec
-0.05
10
n
20
-0.05
0
10
n
20
0
500
t in µs e c
sampling interval T
sampling frequency fs= 1/T
3
Digital signals: time representations
0.5
x(n)
8000 samples
100 samples
line with dots
0
-0.5
0
1000
2000
3000
4000
5000
6000
7000
8000
x(n)
0.5
0
-0.5
x(n)
vertical quantization
integer
e.g. -32768 .. 32767
normalized
e.g. -1 .. (1-Q)
Q = quantization step
0
100
200
300
400
500
600
700
800
900
1000
0
10
20
30
40
50
60
70
80
90
100
0.05
0
-0.05
n
4
Spectrum: analog vs. digital signal
sampling leads to a replication of the baseband spectrum
5
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
6
Discrete Fourier Transform
Magnitude
Phase
7
Discrete Fourier Transform (example)
FFT with 16 points
cosine (16 points)
1
Cos ine s ignal x(n)
a)
0
-1
0
4
6
b)
8
10
12
14
16
n
1
Ma gnitude s pe ctrum |X(k)|
0.5
0
0
2
4
6
8
10
12
14
16
k
1
c)
magnitude (16 points)
normalization:
0 dB for sinusoid ±1
magnitude
(frequency points)
2
Ma gnitude s pe ctrum |X(f)|
0.5
kfs / N
0
0
step fs/N
0.5
1
1.5
2
f in Hz
2.5
3
3.5
x 10
4
20
magnitude dB vs. Hz
|X(f)| in dB
Magnitude s pe ctrum |X(f)| in dB
0
-20
-40
0
0.5
1
1.5
2
f in Hz
2.5
3
3.5
x 10
8
4
Inverse Discrete Fourier Transform (DFT)
if X(k) = X*(N-k)
then IDFT gives N discrete-time real values x(n)
9
Frequency resolution
Zero padding: to increase
frequency resolution
8 s amples
8-point FFT
10
2
8
6
|X(k)|
x(n)
1
0
4
2
-1
0
0
2
4
6
0
8 s amples + ze ro-pa dding
2
4
6
16-point FFT
10
2
8
6
|X(k)|
x(n)
1
0
4
2
-1
0
0
5
10
n
15
0
5
10
15
k
10
Window functions
to reduce leakage:
weight audio samples by
a window
Hamming window
wH(n) = 0.54 – 0.46 cos(2 n/N)
Blackman window
wB(n) = 0.42 – 0.5cos(2 n/N) + 0.08(4 n/N)
11
Window
Reduction of the leakage effect by
window functions:
(a) the original signal,
(b) the Blackman window function
of length N =8,
(c) product x(n)w(n) with 0 n N-i,
(d) zero-padding applied to z(n)
w(n) up to length N = 16
The corresponding spectra are
shown on the right side.
12
Spectrograms
13
Waterfall representation
S ignal x(n)
1
0.5
x(n)
0
-0.5
-1
Magnitude in dB
0
1000
2000
3000
4000
5000
6000
7000
n
Waterfall Repres entation of S hort-time FFTs
8000
0
-50
0
2000
4000
6000
-100
0
5
10
15
20
n
f in Hz
14
Digital systems
15
Definitions
Unit impulse
Impulse reponse h(n) = output to a unit impulse
h(n) describes the digital sistem
Discrete convolution:
y(n)=x(n)*h(n)
16
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)
17
Simple digital system
weighted sum over several input samples
18
Transforms
Frequency domain desciption of the digital system
Z transform
Discrete time Fourier
transform
Transfer function H(z):
Z transform of h(n)
Frequency response:
Discrete time Fourier
transform of h(n)
19
Causal and stable systems
Causality: a discrete-time system is causal
if the output signal y(n) = 0 for n <0 for a given input signal
u(n) = 0 for n <0.
This means that the system cannot react to an input before the
input is applied to the system
Stability: a digital system is stable if
stability implies that transfer function H(z) and frequency
response are related by
20
IIR systems
= system with infinte impulse response
e.g. second order IIR system
Difference equation
Transfer function
21
IIR systems
= system with infinte impulse response h(n)
Difference equation
Z transform of diff. eq.
Transfer function
22
FIR systems
= system with finite impulse response h(n)
e.g. second order FIR system
Difference equation
Z transform of diff. eq.
Transfer function
23
Fir example
computation of frequency response
(a ) Impuls e Re s pons e h(n)
(b) Magnitude Res pons e |H(f)|
0.8
0.6
0.1
0.4
|H(f)|
0.2
0
0.2
-0.1
0
0
2
4
0
n
(c ) P ole/Zero plot
10
20
30
40
f in kHz
(d) P ha s e Res pons e
H(f)
0
1
-0.5
4
0
H(f)/
Im(z)
impulse response
magnitude resp.
pole/zero plot
phase resp.
0.3
-1
-1.5
-1
-2
-1
0
1
Re (z)
2
0
10
20
f in kHz
30
40
24