Course pages 2017–18

# Digital Signal Processing

**Principal lecturer:** Dr Markus Kuhn**Taken by:** Part II**Past exam questions**

No. of lectures: 12

Suggested hours of supervisions: 3

Prerequisite courses: Mathematical Methods I and III from the NST Mathematics course, Mathematical Methods for Computer Science, LaTeX and MATLAB (recommended).

## Aims

This course teaches the basic signal-processing principles necessary to understand many modern high-tech systems, with digital-communications examples. Students will gain practical experience from numerical experiments in MATLAB-based programming assignments.

## Lectures

**Signals and systems.**Discrete sequences and systems, their types and properties. Linear time-invariant systems, convolution.**Phasors.**Eigen functions of linear time-invariant systems. Review of complex arithmetic. Some examples from electronics, optics and acoustics.**Fourier transform.**Phasors as orthogonal base functions. Forms and properties of the Fourier transform. Convolution theorem.**Dirac’s delta function.**Fourier representation of sine waves, impulse combs in the time and frequency domain.**Discrete sequences and spectra.**Periodic sampling of continuous signals, periodic signals, aliasing, interpolation, sampling and reconstruction of low-pass and band-pass signals, spectral inversion.**Digital modulation.**IQ representation of band-pass signals, in particular AM, FM, PSK, and QAM signals.**Discrete Fourier transform.**Continuous*versus*discrete Fourier transform, symmetry, linearity, review of the FFT, real-valued FFT.**Spectral estimation.**Short-time Fourier transform, leakage and scalloping phenomena, windowing, zero padding.**Finite impulse-response filters.**Properties of filters, implementation forms, window-based FIR design, use of frequency-inversion to obtain high-pass filters, use of modulation to obtain band-pass filters, FFT-based convolution.**Infinite impulse-response filters.**Sequences as polynomials,*z*-transform, zeros and poles, some analog IIR design techniques (Butterworth, Chebyshev I/II, elliptic filters).**Random sequences and noise.**Random variables, stationary processes, autocorrelation, crosscorrelation, deterministic crosscorrelation sequences, filtered random sequences, white noise, exponential averaging.**Correlation coding.**Random vectors, dependence*versus*correlation, covariance, decorrelation, matrix diagonalization, eigen decomposition, Karhunen-Loève transform, principal component analysis. Relation to orthogonal transform coding using fixed basis vectors, such as DCT.

## Objectives

By the end of the course students should be able to

- apply basic properties of time-invariant linear systems;
- understand sampling, aliasing, convolution, filtering, the pitfalls of spectral estimation;
- explain the above in time and frequency domain representations;
- use filter-design software;
- visualize and discuss digital filters in the
*z*-domain; - use the FFT for convolution, deconvolution, filtering;
- implement, apply and evaluate simple DSP applications in MATLAB;
- apply transforms that reduce correlation between several signal sources;
- explain the basic principles of some widely-used modulation and image-coding techniques.

## Recommended reading

* Lyons, R.G. (2010). *Understanding digital signal processing.* Prentice Hall (3rd ed.).

Oppenheim, A.V. & Schafer, R.W. (2007). *Discrete-time digital signal processing.* Prentice Hall (3rd ed.).