ES97H Biomedical Signal Processing 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.

BIOMEDICAL SIGNAL ANALYSIS

system H(s) Systems Analysis. Block Diagrams. Transfer function Simple Input Signal Output Signal system H(s)

H1(s) H2(s) Systems Analysis. Block Diagrams. Transfer function Series System consists of two sub-systems H1(s) H2(s) Sub-system 1 Sub-system 2 Block diagram

H1(s) H2(s) Systems Analysis. Block Diagrams. Transfer function Cascade Connection System consists of two sub-systems H1(s) + Sub-system 1 H2(s) - Sub-system 2 Block diagram

Transfer function. Poles and Zeros A transfer function H(s) can always be written as a rational function of s, that is as a ratio of two polynomials Polynomials can be factorized as follows The are called the zeros of H(s) The are called the poles of H(s) Poles define the stability of the system; if Re(pi)<0 LTI is stable

system Transfer function. Impulse response Consider a test signal as the delta-function, d(t). Since Consequently, if x(t)=d(t) then X(s)=1 and H(s)=Y(s) The output signal y(t)=h(t) corresponds to the impulse response Dream: the transfer function H(s) of a system can be obtained by applying an impulse, x(t)=d(t), (whilst the system is in its quiescent state) to the system and measuring its response, y(t). H(s) is then given by the Laplace transform of y(t). Input Signal Output Signal system

Transfer function. Frequency Response Select the test signal in the form of harmonic functions: Such signals can be realized experimentally Substitute s=jw for the transfer function, then the frequency response is That is, if we know H(s) we can obtain H(w) by replacing s by jw Note: it is assumed that the system is in a stable steady state.

Transfer function. Frequency Response The frequency response H(jw) is complex function of w Therefore the polar form is used 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!

Systems Response to a harmonic signal. A cosine input If a signal Acos(w0t) is applied to a system with transfer function, H(s), the response is still a cosine but with an amplitude and phase Note. We don’t need to use inverse Laplace Transform to estimate the response in time domain. the system response to

Systems Response to a harmonic signal. A cosine input Example. Input signal Transfer function Output signal y(t) - ? Rewrite the transfer function as the frequency response in the polar form Gain Phase Calculate the gain and phase for w =2p

Fourier Analysis. Spectrum Most signals encountered in engineering can be represented in both the time and the frequency domains. These representations are uniquely related. So, alternating the signal in one domain will alter its representation in the other domain as well. Time domain. The plot of a signal, x(t), as a function of time, t. Frequency domain. Spectrum. Amplitude, A1, and phase, f1, of harmonic signal(s) of a given frequency, f 1or w1. A f, (radian) 2p A1 f1 f, (Hz) f, (Hz)

Fourier Analysis. Amplitude response and phase response Example. Time domain. The signal is a sum of a number of harmonic functions. Frequency domain. Amplitude spectrum Phase spectrum phase, f amplitude, A w, (rad./s) angular frequency, w, (rad./s)

Fourier Analysis. Oscillations Example. Carcadian rhythm Many species, from bacteria to humans, maintain a daily rhythm of life by way of a circadian clock. Body temperature ,Tb, and oxygen consumption, V02, from a chronically instrumented pigeon over a 48-hour period .

If the signal is periodic then Fourier series are used. A periodic signal has the property x(t) = x(t+T), T is the fundamental period, f0=1/T or w0 = 2p/T is the fundamental frequency. Trigonometric form of signal a0, an, bn are the Fourier coefficients T is the fundamental period w0 is the fundamental frequency For spectrum we need to calculate the coefficients

Fourier series of periodic signals Alternative representation of Fourier series a0, An, fn are the Fourier coefficients a0 is the DC component, mean value of the signal Frequency components occurs at frequencies of nw0 and they are characterized by amplitudes An and phases fn. The n=1 term is called the fundamental frequency component and the n=2,3,... components are called 2nd, 3rd,... harmonics respectively. It is an odd harmonic if n is odd and an even harmonic if n is even. Plots An versus w and fn versus w are amplitude and phase spectra of x(t) respectively.

Fourier Series. Parseval’s Theorem The “power”, P, of a periodic signal, x(t), can be written via Fourier coefficients an and bn, Time domain Frequency domain The power (energy) in the time domain and the frequency domain are equal. Note, T is the duration of the signal here.

Fourier Series Example. Spectrum of periodic pulse train (the square wave). +A x(t) t Fourier series contains infinite number of terms. So the spectrum contains infinite number of peaks. T -A

Fourier Series Example. Spectrum of periodic pulse train (the square wave). Gibbs Phenomenon. Approximation of square wave by 125 harmonics

Fourier Series. Complex representations Complex form. cn are complex coefficients connected with the previously used coefficients as Parseval’s theorem: Example. The spectrum of the square wave.

Fourier series representation T is the period of pulses T1 is the pulse duration Example. Pulse train. Fourier series representation Complex Fourier coefficients The sinc function

Complex Fourier coefficients T is the period of pulses T1 is the pulse duration Example. Pulse train. Complex Fourier coefficients The amplitude spectrum As the period T tends to infinity number of frequency component that occur in the frequency interval [0,w0] tends to infinity too. |cn| So, if we consider one pulse (impulse) only, the Fourier series are useless . |cn|

Fourier transform of aperiodic signals. Fourier transform of x(t) is denoted X(w). It is a continuous function of the frequency. The inverse Fourier transform is Fourier pairs is the magnitude or amplitude spectrum is the phase spectrum Frequency response Laplace transform Fourier transform: Parseval’s theorem: Energy of an aperiodic signal Energy (Power) spectrum

Fourier transform Example. Single rectangular pulse The amplitude spectrum The phase spectrum

Fourier transform Example. Single exponential pulse The amplitude spectrum The phase spectrum

Fourier transform Example. The unit impulse function (delta function) d(t) White spectrum The amplitude spectrum The phase spectrum Example. Harmonic functions

Fourier transform Example. Harmonic functions

Fourier transform of periodic and aperiodic signals Spectrum is a continuous function of the frequency Periodic signals Spectrum of the periodic signal The period of the signal Spectrum is discrete, specified in (in)finite number of points only. A general signal can have both aperiodic and periodic components, consequently the spectrum will have both continuous and discrete components:

Fourier transform. Effects of a finite-duration of signal. Edge effect Consider a harmonic signal, y(t), of a finite duration, T. The product (multiplication) in the time domain corresponds to the convolution in the frequency domain and vice versa. The discrete spectrum is transformed to a continuous one