# Digital Signal Processing

**Principal lecturer:** Dr Ali Ozgur Yontem

**Taken by:** Part II CST 75%

**Term:** Michaelmas

**Hours:** 16

**Format:** In-person lectures

**Class limit:** max. 24 students

**Prerequisites:** Mathematical Methods I and III from the NST Mathematics course (or equivalent), LaTeX and MATLAB (recommended)

**timetable**

## Aims

This course teaches the basic signal-processing principles necessary to understand many modern high-tech systems, with application examples focussing on audio processing, image coding, communication systems and software-defined radio. Students will gain practical experience from numerical experiments in programming assignments (in MATLAB, NumPy or Julia).

## Lectures

**Signals and systems.**Discrete sequences and systems: types and properties. Amplitude, phase, frequency, modulation, decibels, root-mean square. Linear time-invariant systems, convolution. Some examples from electronics, optics and acoustics.**Phasors.**Eigen functions of linear time-invariant systems. Review of complex arithmetic. Phasors as orthogonal base functions.**Fourier transform.**Forms and properties of the Fourier transform. Convolution theorem. Rect and sinc.**Dirac’s delta function.**Fourier representation of sine waves, impulse combs in the time and frequency domain. Amplitude-modulation in the frequency domain.**Discrete sequences and spectra.**Sampling of continuous signals, periodic signals, aliasing, interpolation, sampling and reconstruction, sample-rate conversion, oversampling, spectral inversion.**Discrete Fourier transform.**Continuous*versus*discrete Fourier transform, symmetry, linearity, FFT, real-valued FFT, FFT-based convolution, zero padding, FFT-based resampling, deconvolution exercise.**Spectral estimation.**Short-time Fourier transform, leakage and scalloping phenomena, windowing, zero padding. Audio and voice examples. DTFM exercise.**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.**Infinite impulse-response filters.**Sequences as polynomials,*z*-transform, zeros and poles, some analog IIR design techniques (Butterworth, Chebyshev I/II, elliptic filters, second-order cascade form).**Band-pass signals.**Band-pass sampling and reconstruction, IQ up and down conversion, superheterodyne receivers, software-defined radio front-ends, IQ representation of AM and FM signals and their demodulation.**Digital communication.**Pulse-amplitude modulation. Matched-filter detector. Pulse shapes, inter-symbol interference, equalization. IQ representation of ASK, BSK, PSK, QAM and FSK signals. [2 hours]**Random sequences and noise.**Random variables, stationary and ergodic processes, autocorrelation, cross-correlation, deterministic cross-correlation sequences, filtered random sequences, white noise, periodic averaging.**Correlation coding.**Entropy, delta coding, linear prediction, dependence*versus*correlation, random vectors, covariance, decorrelation, matrix diagonalization, eigen decomposition, Karhunen-Loève transform, principal component analysis. Relation to orthogonal transform coding using fixed basis vectors, such as DCT.**Lossy versus lossless compression.**What information is discarded by human senses and can be eliminated by encoders? Perceptual scales, audio masking, spatial resolution, colour coordinates, some demonstration experiments.**Quantization, image coding standards.**Uniform and logarithmic quantization, A/µ-law coding, dithering, JPEG.

## Objectives

- 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;
- familiarity with a number of signal-processing concepts used in digital communication systems

## Assessment - Part II students

- Three homework programming assignments, each comprising 20% of the mark
- Written take-home assignment, comprising 40% of the total mark.

## Recommended reading

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

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

Stein, J. (2000). *Digital signal processing – a computer
science perspective.* Wiley.