Digital Signal Processing

Aims

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

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. Eigenfunctions 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. DTMF 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, BPSK, 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, eigendecomposition, 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, and 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;
• be familiar with a number of signal-processing concepts used in digital communication systems.

Recommended reading

Lyons, R.G. (2010). Understanding digital signal processing. Prentice Hall (3rd ed.).
Oppenheim, A.V. and Schafer, R.W. (2009). Discrete-time signal processing. Prentice Hall (3rd ed.).
Stein, J. (2000). Digital signal processing – a computer science perspective. Wiley.

Class size

This module can accommodate a maximum of 24 students (16 Part II students and 8 MPhil students).

Assessment – Part II students

• Three homework programming assignments, each comprising 1/6 of the total mark.
• Written test, comprising 1/2 of the total mark.

The assessments listed for this module may be subject to change following the outcome of the consultation on the addition of vivas to Module assessment modes.