systems (filters)
Post on 30-Dec-2015
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Non-periodic signal has continuous spectrum
Sampling in one domain implies periodicity in another domain
time frequency
Periodic sampled signal has always discrete and periodic spectrum
Linear system
k*input
system
k*output
Frequency response
input
system
output
frequency response = output/input
Output at time t depends only on the input at time t
Memoryless system (amplifier)
Frequency response of the system
Magnitude (dB)
30
phase
frequency frequency1 10 100 1000 1 10 100 1000
2x
System with a memory (differentiator)
Frequency response of the differentiator (high-pass filter)
time0 t0
in
0 t0
time
out
1 sampledelay
-
System with a memory (integrator)
Frequency response of the integrator (low-pass filter)
time0 t0
in
0 t0
time
out
1 sampledelay
+
delay TD
-TD
e.t.c
TD=T1
TD=3T2
TD=5T3
Comb filter
1/TD 3/TD 5/TD
1
Frequency response of the system
magnitude
0
e.t.c.
frequency
const
T
€
∞ For a single pulse,• the period becomes infinite• the sum in Fourier series becomes integral
THE LINE SPECTRUM BECOMES CONTINUOUS
timet 0€
∞
frequencyt
€
∞
system
Dirac impulse Impulse response
time time
Fouriertransform
frequency
Frequency response
Dirac impulse contains all frequencies
Fourier transform of the impulse response of a system is its frequency response!
Sinusoidal signal (pure tone)
€
∞
€
−∞T
time [s] frequency [Hz]
1/T
Its spectrum
?Truncated sinusoidal signal Its spectrum
time [s]
Truncated signal
Infinite signal
multiplied by
square window
Multiplication in one (time) domain is convolution in the dual (frequency) domain
10 ms 2 ms
Pulse train
Its magnitude spectrum
f = 1/2 103 =500 Hz
line spectrum with |sinc| envelope
1/tp 2/tp 3/tp
frequency
0
continuous |sinc| function
tp
∞∞-
Convolution of the impulse with any function yields this function
frequency [Hz]
1000
Spectrum of an infinite 1 kHz sinusoidal signal
Truncated
Narrow-band(high frequency resolution)
system
Wide-band(low frequency resolution)
system
frequency
time
Narrow-band (high frequency resolution) Broad-band (low frequency resolution)
Long impulse response (low temporal resolution)
Short impulse response (high temporal resolution)
Time-Frequency Compromise
• Fine resolution in one domain (f-> 0 or t->0) requires infinite observation interval and therefore pure resolution in the dual domain (-> or F-> )– 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
T
to analysis the signal appear to be periods with the period T
T
Discrete Fourier Transform
1
0
21 )()(N
n
N
knj
N ekXnx
1
0
21 )()(N
n
N
knj
N enxkX
Discrete and periodic in both domains (time and frequency)
Analysis window 50 ms
time [s]0 1.2
Analysis window 5 ms
time [s]0 1.2
freq
uenc
y [k
Hz]
5
0
frequency
log amplitude
frequency
time [s]01.2fr
eque
ncy
[kH
z]
5
0
€
Sn (ejω ) = s(m) ⋅w(n −m)e− jmω
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.
Short-term discrete Fourier transform
)()()( mnwemseSm
jmjn
)()(
terms twoof
nconvolutio represents aboveequation the
),frequency particular a(at fixed is if
0
0
mwems mj
mje 0
W(m))( jeS)(ms
w(m) window theof` )(
responsefrequency by given shapefilter theand
frequency center h filter wit pass-band aby
filteringlinear representsn convolutio The
0
W
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