1 transfer function. frequency response the frequency response h(j ) is complex function of ...
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Transfer function. Frequency Response
The frequency response H(j) is complex function of Therefore the polar form is used
( )( ) ( ) j HH H e
( )
( ) ( )
H
H
is the modulus (gain), the ratio of the amplitudes of the output and the input;
is the phase shift between the output and the input.
Thus, the frequency response is fully specified by the gain and phase over the entire range of frequencies
Both gain and phase are experimentally accessible!
[0, )
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Systems Response to a harmonic signal. A cosine input
If a signal Acos(0t) is applied to a system with transfer function, H(s), the response is still a cosine but with an amplitude and phase
0( )A H
00 ( )t H
( )x t ( )y t( )H s
Note. We don’t need to use inverse Laplace Transform to estimate the response in time domain.
0 0 0( ) ( ) cos ( )y t A H j t H j the system response to 0( ) cosx t A t
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Fourier transform of periodic and aperiodic signals
0( ) 2 ( )nn
X c n
Aperiodic signals
Periodic signal
0
2T
( ) ( )x t x t T
Fourier Series Spectrum(discrete)
0
0
0
0
( )
1( )
jn tn
n
t Tjn t
n
t
x t c e
c x t e dtT
Fourier Transform Spectrum(continuous)
( )X ( ) ( ) j tX x t e dt
( )x t
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Effects of a finite-duration of signal. Edge effect
Consider a harmonic signal, y(t), of a finite duration, T.( ) sin( )x t t { ( )} { ( )} { ( )}y t x t v t F F F
1, / 2( )
0, / 2
t Tv t
t T
( ) ( ) ( )y t x t v t
1{ ( )} ( ) ( )
2y t X V d
F
The product (multiplication) in the time domain corresponds to the convolution in the frequency domain and vice versa.
( ) sin( )x t a t
( )v t
( ) ( ) ( )y t x t v t
( )X ( )V sin ( ) / 2 sin ( ) / 2
( )2 ( ) / 2 ( ) / 2
T TjTY
T T
( ) sinc ( ) / 2 sinc ( ) / 22
jTY T T
The discrete spectrum is transformed to a continuous one
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Spectral representation of random (stochastic) signals
Stochastic signal x(t) means non-regular, non-periodic, non-deterministic, non-predictable etc.
Stochastic signal is a realization of a stochastic process
We need statistical measures to describe the stochastic process
Power Spectrum or Power Spectral Density (PSD) Sxx estimates how the total power (energy) is distributed over frequency.
)( ) (xj t
x xxS R e d
0
1) lim ( )( ( )
T
xxT
x t x dT
R t t
Auto-correlation function
Julius S. Bendat, Allan G. PiersolRandom data : analysis and measurement procedures (e-book)http://encore.lib.warwick.ac.uk/iii/encore/record/C__Rb2636504
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Spectral representation of random (stochastic) signals
Power Spectrum Density is a function expressed as a power value (Signal Units)2 per unit frequency range (Hz) SU2/Hz.
( )x t
(0)xxS
1( )xxS f
2( )xxS f
1( )xx NS f
Band-Passfilters
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Spectrum of stochastic signal
Fourier transform is linear operation, so the transform for stochastic realizations will be also stochastic and vary form one realization to other.
Statistical measures:Mean valueDispersionDistribution
Task: to define properties of the process, not a single realization. Solution is the use of
Power Spectrum or Spectral Density (assume stationarity)
2
2
1
( ) 1( ) ( )
2 2limN
xx iT i
E XS X
T TN
It is not the amplitude spectrumThere is no the phase spectrumThere is no an inverse transform
1( )X
2 ( )X
( )nX
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Lecturer: Dr Igor Khovanov
Office: D207
[email protected] Syllabus:
Biomedical Signal Processing. Examples of signals. Linear System Analysis. Laplace Transform. Transfer Function.
Frequency Response. Fourier Transform. Discrete Signal Analysis. Digital (discrete-time) systems. Z-transform.
Filtering. Digital Filters design and application.
Case Study.
ES97H Biomedical Signal Processing
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Discrete-time signals
Many biomedical measurements such as arterial blood pressure are inherently defined at all instants of time. The signals resulting from these measurements are continuous-time signals x(t)
Within the biomedical realm one can consider the amount of blood ejected from the heart with each beat as a discrete-time variable,and its representation as a function of a time variable or beat number (which assumes only integer values and increments by one with each heartbeat) would constitute a discrete-time signal xi (x[i]).
Discrete-time signals can arise from inherently discrete processes as well as from sampling of continuous-time signals.
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Discretization. Sampling
Analogue-Digital-Converter (ADC) is used for sampling and quantization (digitization) of a continuous signal, x(t).
( ) ix t x
sampling
quantizationADC levels
it
( ) [ ] INT ii i i
xx t x i x c
x
x
( )i i ix t c
i Quantization error (noise)
Note we will ignore the errors i further and concentrate on sampling
0
( ) ( ) ( )i ix t x x t d
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Sampling
Continuous time signal
Discrete time signal
Discrete time signal
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Fourier transform of discrete-time signal. Theoretical analysis
0
( ) ( ) ( ) ( )i ix t x iT x x iT d
Consider a discrete-time signal {xi}, an infinite sequence, 1,0,1ix i obtaining by sampling with sampling time T from
a continuous-time signal x(t).
{ } ( ) ( ) ( )ii
x x iT t iT x t TS
TS is the time sampling function
{ } { ( ) } { ( )} { } ( )i Sk
x x t TS x t TS X k
F F F F
The Fourier transform of the discrete-time signal is the convolution of the Fourier transforms of the continuous-time signal, X(), and TS.
Angular sampling frequency
2S T
The spectrum of the D-T signal repeats periodically the spectrum of C-T signal with intervals S
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Fourier transform of discrete-time signal. Theoretical analysis
Here the symbol denotes the convolutionF/2
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Fourier transform of discrete-time signal. Theoretical analysis
The Fourier transform of the discrete-time signal is a continuous function of frequency
The inverse Fourier transform of the discrete-time signal has finite limits of integration
( ) j iTi
i
X T x e
1
( )2
N
N
j iTix X e d
N
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2 2
SN
SN
ff
T
Nyquist frequency defines the maximal frequency in the spectrum of the discrete-time signal.
Kotel’nikov-Nyquist-Shannon theorem. A band-limited continuous signal that has been sampled can be perfectly reconstructed from an infinite sequence of samples (discrete-time signal) if the sampling rate exceeds 2fm samples per second, where fm is the highest frequency in the original signal.
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Aliasing effect. Frequency ambiguity
Corollaries of the Kotel’nikov theorem.
Harmonic signals having frequencies 2kfN f are indistinguishable when sampled with rate 2fN.f is an arbitrary
Aliasing refers to an effect that causes different signals to become indistinguishable
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Aliasing effect. Artifacts and distortions
Corollaries of the Kotel’nikov theorem.
The spectrum of signal sampled with 4fN
X(f ) X(f )
fN fN
Reflection of spectral parts
The spectrum of signal sampled with 2fN
2fN
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Sampling. Anti-aliasing (analogue) filters
The scheme for signal sampling:
Analog(ue) means non-digital
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The discrete Fourier transform. Digital signal processing
Consider a discrete-time signal (series) {xi} of a finite length, N
0,1, 1ix i N obtaining by sampling with sampling time T from
a continuous-time signal x(t).
12 /
0
{ } ( )N
j ik Ni k k i
i
x X f X T x e
F
The discrete Fourier transform of the discrete-time signal is
a finite length, N, sequence of complex coefficient Xk.
The inverse discrete Fourier transform of the discrete-time signal is1
1 2 /
0
1{ }
Nj ik N
k i kk
X x X eNT
F
k
kf
NT 0,1, / 2 1k
kf k N
NT Frequency correspond 0 to fN
/ 2, / 2 1,k
kf k N N N
NT Frequency correspond 0 to fN
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The discrete Fourier transform. Digital signal processing
Time series {xi} i=0...N1. Time sampling T
Fourier transform Xk , k=0...N
, 0, / 2 1k
kf k N
NT Frequency series
Amplitude spectrum series
Phase spectrum series
, 0, / 2 1kX k N
1 Im( )tan , 0, / 2 1
Re( )k
kk
XX k N
X
Spectral resolution
1f
NT
Parseval’s theorem (!). 21 1
22
0 0
1
( )
N N
i ki k
x XNT
For calculations one uses Fast Fourier Transform (FFT), typically with N=2m , m is an integer.
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Fourier transform. Edge Effect
Consider a harmonic signal, y(t), of a finite duration, P.( ) sin( )x t t { ( )} { ( )} { ( )}y t x t v t F F F
1, / 2( )
0, / 2
t Pv t
t P
( ) ( ) ( )y t x t v t
1{ ( )} ( ) ( )
2y t X V d
F
The product (multiplication) in the time domain corresponds to the convolution in the frequency domain and vice versa.
( ) sin( )x t a t
( )v t
( ) ( ) ( )y t x t v t
( )X ( )V sin ( ) / 2 sin ( ) / 2
( )2 ( ) / 2 ( ) / 2
P PjTY
P P
( ) sinc ( ) / 2 sinc ( ) / 22
jTY P P
The discrete spectrum is transformed to a continuous one
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Avoidance of the edge effect for periodic signal
Consider a harmonic discrete-time signal 0cos( ) 1, 2mix A iT i N
Select the sampling period (rate) as0
2 lT
N
l=1,2,3...
That is series of duration PNT contains an integer number, l, of signal periods
Then the expression for the Fourier transform of continuous signal
0 0
0 0
sin ( ) / 2 sin ( ) / 2( )
2 ( ) / 2 ( ) / 2
P PAPX
P P
has the following form
0 0
0 0
sin ( ) / 2 sin ( ) / 2( )
2 ( ) / 2 ( ) / 2k k
kk k
P PAPX
P P
and by replacing
2k
k
NT
Finally, arrive to the expression, that specifies two peaks localized on signal frequency
/ 2,( )
0,k
ANT k lX
k l
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Avoidance of the edge effect for periodic signal
Consider a harmonic discrete-time signal 0cos( ) 1, 2mix A iT i N
Amplitude spectrum Amplitude spectrum
( )kX ( )kX
0 00 0
0
2P TN l
Signal duration Signal duration0
2aP TN l l
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The discrete-time series. Some comments on digital processing
Start with the continuous-time signal x(t) of a finite duration, P
Sampling with rate fS=1/T (skipping quantization) leads to
a sequence {xi}, i=1,...N; where N=P/T
The sequence has the sampling period Tcomputer=1 and sampling frequency fcomputer=1, it means that fN=0.5. It is true for any data in computer!
So we develop approaches for computer data and then go back to “real” signal for interpretation.
00 0
00
1( ) ( ) cos( ) cos 2 cos 2 cos 2i i comp
S S
comp comp SS
fx t x t x A iT A f i A i A f i
f f
ff f f f
f