Systems
(filters)
Non-periodic signal has
continuous spectrum
Sampling in one domain
implies periodicity in
another domain
Periodic sampled signal has always discrete and periodic spectrum
time
frequency
PROCESSING
One way of “signal processing”
Linear system
k*input
k*output
system
frequency response = output/input
Frequency response
input
output
system
deciBel [dB]
20 log
Output ( f )
Input ( f )
Log-log frequency response

Memoryless system (amplifier)
2x
Output at time t depends only on the input at time t
Frequency response
of the system
Magnitude (dB)
3
phase
0
frequency
1 10 100 1000
1 10 100 1000
frequency
System with a memory (differentiator)
in
Frequency response of the differentiator (high-pass filter)
0
t0
time
1 sample
delay
out
0
t0
time
System with a memory (integrator)
in
Frequency response of the integrator (low-pass filter)
0
t0
time
+
1 sample
delay
out
0
t0
time
TD
const
delay TD
Comb filter
Frequency response
of the system
TD=T1
TD=3T2
magnitude
TD=5T3
1
e.t.c
e.t.c.
0
frequency
1/TD 3/TD 5/TD
linear system
nonlinear system
output
output
input
input
noisy system
noise
Pulse train
10 ms
2 ms
Its magnitude spectrum
10 ms
2 ms
20 ms
T


For a single pulse,
• the period becomes
infinite
• the sum in Fourier
series becomes
integral
THE LINE SPECTRUM
BECOMES
CONTINUOUS


dt
Dirac impulse
Dirac impulse contains
all frequencies
0
time

1/dt

Impulse response
Frequency response
Fourier
transform
system
time
frequency
time
frequency
Fourier transform of the impulse response of a system is its frequency response!
Sinusoidal signal (pure tone)

Its spectrum
T

1/T

Truncated sinusoidal signal
time [s]
frequency [Hz]
Its spectrum
DT
?
Truncated signal
time [s]
Infinite signal
multiplied by
square window
Multiplication in one (time) domain is convolution in the dual (frequency) domain
tp
Pulse train
10 ms
2 ms
-∞
∞
Its magnitude spectrum
0
f = 1/2 103 =500 Hz
line spectrum with |sinc| envelope
1/tp
2/tp
3/tp
frequency
continuous |sinc| function
Convolution of the impulse with any function yields this function
Spectrum of an infinite
1 kHz sinusoidal signal
1000
frequency [Hz]
Truncated
Dt = ∞
Dt = 100 ms
Dt = 13 ms
0
850 Hz
Narrow-band
Wide-band
(high frequency resolution) (low frequency resolution)
system
system
frequency
time
Narrow-band (high frequency resolution)
Long impulse response
(low temporal resolution)
Broad-band (low frequency resolution)
Short impulse response
(high temporal resolution)

Time-Frequency Compromise
• Fine resolution in one domain (df-> 0 or dt->0)
requires infinite observation interval and
therefore pure resolution in the dual domain
(DT->  or DF->  )
– You cannot simultaneously know the exact
frequency and the exact temporal locality of the
event

– infinitely sharp (ideal) filter would require
infinitely long delay before it delivers the output
signal is typically changing in time (non-stationary)
time
short-term analysis: consider only a short segment of the signal at any given time
DT
DT
to analysis the signal appear to be periods with the period DT
Non-stationary turns into periodic
Discrete Fourier Transform
x(n) 
N 1
1
N
 X (k )  e
n 0
j
2kn
N
X (k ) 
N 1
1
N
 x(n)  e
j
2kn
N
n 0
Discrete and periodic in both domains (time and frequency)
Short-term Discrete Fourier Transform
Signal multiplied by the window
Spectrum of the signal convolves with the spectrum of the window
frequency
time
time
time
frequency
Analysis window 5 ms
Analysis window 50 ms
frequency [kHz]
5
0
0
time [s]
1.2
0
time [s]
log amplitude
frequency
frequency
1.2
log amplitude
time [s]
frequency
frequency [Hz]
frequency [kHz]
4
0
0
time [s]
/a;/
/e:/
/i:/
6
/o:/
/u:/
Speech production
/j/
/u/
/ar/
/j/
/o/
/j/
/o/
j
Sn (e ) 

 s(m)  w(n  m)e
 jm
m
Fourier transform of the signal s(m) multiplied by the window w(n-m)
Spectrum is the line spectrum of the signal convolved with the
spectrum of the window

frequency [kHz]
Spectral resolution of
the short-term Fourier
analysis is the same at
all frequencies.
5
0
0
1.2
time [s]
Short-term discrete Fourier transform
Sn (e j ) 

 jm
s
(
m
)
e
 w(n  m)

m  
e j 0 m
if  is fixed (at a particularfrequency0 ),
theequation above representsconvolution
of two terms
s(m)  e  j 0 m  w(m)
T heconvolution representslinear filtering
by a band - pass filter with centerfrequency0
and thefiltershapegiven by frequencyresponse
W ( ) of`the window w(m)
s (m)
W(m)
S (e j )
Homework