Course pages 2019–20

**Subsections**

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Unit: Digital Signal Processing

The whole course is for Part II 75% students, while Part II 50% students can take one question in Paper 7 about part 1 of this course.

*Lecturer: Dr M.G. Kuhn*

*No. of lectures:* 16

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

### Aims

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

### Lectures

Part 1 (about 9-10 lectures) focusses on basic theory and audio applications.

**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).

Part 2 (about 6-7 lectures) adds material on software-defined radio techniques, statistical signals, audio-visual signal compression and linear feedback control systems.

**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. Clock recovery. Spectral characteristics of binary sequences.[2 hours]**Random sequences and noise.**Random variables, stationary and ergodic processes, autocorrelation, crosscorrelation, 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/mu-law coding, dithering, JPEG.

### Objectives

By the end of part 1 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;

By the end of part 2, students should be able to discuss and explain many fundamental concepts of techniques commonly used in digital communicationsystems in terms of the concepts introduced in part 1.

### 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.).

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

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