1. 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!

2. 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

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

4. 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

5. 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)

6. 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

7. 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

8. ES97H Biomedical Signal Processing
Lecturer: Dr Igor Khovanov
Office: D207
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.

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).
quantization
ADC levels
sampling Dx
Note we will ignore the errors xi further and concentrate on sampling
Quantization error (noise)

11. Sampling
Continuous time signal
Discrete time signal
Discrete time signal

12. 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

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

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
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.
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

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
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

19. 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.

20. 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

21. 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

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

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.