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
One way of “signal processing”
PROCESSING
Linear system
k*input k*output
system
frequency response = output/input
Frequency response
input output
system
Output( f )
deciBel [dB] 20log
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) phase
3
0
frequency frequency
1 10 100 1000 1 10 100 1000
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 TD=T1
of the system
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
Dirac impulse contains
all frequencies
dt 0 time 1/dt frequency
Dirac impulse Impulse response Frequency response
Fourier
system
transform
time time frequency
Fourier transform of the impulse response of a system is its frequency response!
Sinusoidal signal (pure tone) Its spectrum
T
1/T
time [s] frequency [Hz]
Truncated sinusoidal signal 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 1/tp 2/tp 3/tp
f = 1/2 103 =500 Hz frequency
line spectrum with |sinc| envelope continuous |sinc| function
Convolution of the impulse with any function yields this function
Spectrum of an infinite Truncated
1 kHz sinusoidal signal
1000
frequency [Hz]
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) Broad-band (low frequency resolution)
Long impulse response Short impulse response
(low temporal resolution) (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
N 1 2kn N 1 2kn
j
X (k ) e x(n) e
j
x(n) 1
N
N
X (k ) 1
N
N
n 0 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
frequency
time
Analysis window 5 ms Analysis window 50 ms
5
frequency [kHz]
0
0 time [s] 1.2 0 time [s] 1.2
log amplitude
frequency frequency
frequency [Hz]
log amplitude
frequency
time [s]
4
frequency [kHz]
0
0 time [s] 6
/a;/ /e:/ /i:/ /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
Spectral resolution of
the short-term Fourier
analysis is the same at
all frequencies.
5
frequency [kHz]
0
1.2
0 time [s]
t0 time
DT s(f,t0)
fourier spectrum
transform of the short
segment
frequency
time
Short-term discrete Fourier transform
Sn (e j )
m
s(m)e jm w(n m)
if is fixed (at a particular frequency 0 ),
e j 0 m
the equation above representsconvolution s(m) S (e j )
W(m)
of two terms
s (m) e j 0 m w(m)
The convolution representslinear filtering
by a band - pass filter with center frequency 0
and the filter shape given by frequencyresponse
W ( ) of` the window w(m)