revision 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!

revision 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

revision Fourier transform of periodic and aperiodic signals Fourier Series Spectrum (discrete) Periodic signal Fourier Transform Spectrum (continuous) Aperiodic signals

Effects of a finite-duration of signal. Edge effect revision 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

Spectral representation of random (stochastic) signals revision 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(w) estimates how the total power (energy) is distributed over frequency. Auto-correlation function Julius S. Bendat, Allan G. Piersol Random data : analysis and measurement procedures (e-book) http://encore.lib.warwick.ac.uk/iii/encore/record/C__Rb2636504

revision 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. Band-Pass filters

revision 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. Task: to define properties of the process, not a single realization. Solution is the use of Statistical measures: Mean value Dispersion Distribution Power Spectrum or Spectral Density (assume stationarity) It is not the amplitude spectrum There is no the phase spectrum There is no an inverse transform

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.

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.

Discretization. Sampling Analogue-Digital-Converter (ADC) is used for sampling and quantization (digitization) of a continuous signal, x(t). quantization ADC levels sampling Dx Note we will ignore the errors xi further and concentrate on sampling Quantization error (noise)

Sampling Continuous time signal Discrete time signal Discrete time signal

Fourier transform of discrete-time signal. Theoretical analysis Consider a discrete-time signal {xi}, an infinite sequence obtaining by sampling with sampling time T from a continuous-time signal x(t). TS is the time sampling function The Fourier transform of the discrete-time signal is the convolution of the Fourier transforms of the continuous-time signal, X(w), and TS. Angular sampling frequency The spectrum of the D-T signal repeats periodically the spectrum of C-T signal with intervals wS

Fourier transform of discrete-time signal. Theoretical analysis Here the symbol  denotes the convolution

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 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.

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

Aliasing effect. Artifacts and distortions Corollaries of the Kotel’nikov theorem. X(f ) X(f ) Reflection of spectral parts fN 2fN fN The spectrum of signal sampled with 4fN The spectrum of signal sampled with 2fN

Sampling. Anti-aliasing (analogue) filters The scheme for signal sampling: Analog(ue) means non-digital

The discrete Fourier transform. Digital signal processing Consider a discrete-time signal (series) {xi} of a finite length, N obtaining by sampling with sampling time T from a continuous-time signal x(t). The discrete Fourier transform of the discrete-time signal is a finite length, N, sequence of complex coefficient Xk. Frequency correspond 0 to fN Frequency correspond 0 to -fN The inverse discrete Fourier transform of the discrete-time signal is

The discrete Fourier transform. Digital signal processing Time series {xi} i=0...N-1. Time sampling T Fourier transform Xk , k=0...N-1 Spectral resolution Frequency series Amplitude spectrum series Phase spectrum series Parseval’s theorem (!). For calculations one uses Fast Fourier Transform (FFT), typically with N=2m , m is an integer.

Fourier transform. Edge Effect Consider a harmonic signal, y(t), of a finite duration, P. 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

Avoidance of the edge effect for periodic signal Consider a harmonic discrete-time signal Select the sampling period (rate) as l=1,2,3... That is series of duration P=NT contains an integer number, l, of signal periods Then the expression for the Fourier transform of continuous signal has the following form and by replacing Finally, arrive to the expression, that specifies two peaks localized on signal frequency

Avoidance of the edge effect for periodic signal Consider a harmonic discrete-time signal Amplitude spectrum Amplitude spectrum Signal duration Signal duration

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.