1

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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2

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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3

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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4

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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5

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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6

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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7

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

i.khovanov@warwick.ac.uk 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

8

9

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.

10

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

11

Sampling

Continuous time signal

Discrete time signal

Discrete time signal

12

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

13

Fourier transform of discrete-time signal. Theoretical analysis

Here the symbol denotes the convolutionF/2

14

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

21

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.

15

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

16

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

17

Sampling. Anti-aliasing (analogue) filters

The scheme for signal sampling:

Analog(ue) means non-digital

18

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

19

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.

20

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

21

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

22

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

23

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