Systems (filters)

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Systems(filters)
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One way of signal processing
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frequency response output/input
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deciBel dB
Log-log frequency response
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Memoryless system (amplifier)
2x
Output at time t depends only on the input at
time t
Frequency response of the system
Magnitude (dB)
phase
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0
frequency
frequency
1
10
100
1000
1
10
100
1000
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System with a memory (differentiator)
Frequency response of the differentiator
(high-pass filter)
time
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System with a memory (integrator)
Frequency response of the integrator (low-pass
filter)
time
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const
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linear system
nonlinear system
output
output
input
input
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noisy system
noise
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Pulse train
10 ms
2 ms
Its magnitude spectrum
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T

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

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Dirac impulse contains all frequencies
Fourier transform of the impulse response of a
system is its frequency response!
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Sinusoidal signal (pure tone)
T
time s
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Infinite signal
multiplied by
square window
Multiplication in one (time) domain is
convolution in the dual (frequency) domain
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Convolution of the impulse with any function
yields this function
Truncated
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Time-Frequency Compromise

• Fine resolution in one domain (df-gt 0 or dt-gt0)
  requires infinite observation interval and
  therefore pure resolution in the dual domain
  (DT-gt or DF-gt )
• 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
to analysis the signal appear to be periods with
the period DT
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Discrete Fourier Transform
Discrete and periodic in both domains (time and
frequency)
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Short-term Discrete Fourier Transform
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Signal multiplied by the window
Spectrum of the signal convolves with the
spectrum of the window
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time
frequency
time
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Speech production
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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.
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time
t0
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Short-term discrete Fourier transform