Systems(filters)

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

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One way of “signal processing”

Linear system

k*input

system

k*output

Frequency response

input

system

output

frequency response = output/input

deciBel [dB]

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20logOutput( f )

Input( f )

Log-log frequency response

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

linear system

output

input

nonlinear system

output

input

noise

noisy system

10 ms 2 ms

Pulse train

Its magnitude spectrum

10 ms 2 ms 20 ms

T

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∞ For a single pulse,• the period becomes infinite• the sum in Fourier series becomes integral

THE LINE SPECTRUM BECOMES CONTINUOUS

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∞

frequencyt

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∞

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)

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∞

€

−∞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

t = ∞

t = 100 ms

t = 13 ms

0 850 Hz

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

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∞

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

Non-stationary turns into periodic

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)

Short-term Discrete Fourier Transform

Signal multiplied by the window

Spectrum of the signal convolves with the spectrum of the window

time

frequency

time

time

freq

uenc

y

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

freq

uenc

y [H

z]

time [s]

frequency

log

ampl

itude

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freq

uenc

y [k

Hz]

0

time [s]0 6

Speech production

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[kH

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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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