## MAS420 Signal Processing

 Semester 1, 2021/22 10 Credits Lecturer: Dr Gary Verth Timetable Reading List Aims Outcomes Teaching Methods Assessment Full Syllabus

The transmission, reception and extraction of information from signals is an activity of fundamental importance. This course describes the basic concepts and tools underlying the discipline, and relates them to a variety of applications. An essential concept is that a signal can be decomposed into a set of frequencies by means of the Fourier transform. From this grows a very powerful description of how systems respond to input signals. Perhaps the most remarkable result in the course is the celebrated Shannon-Whittaker sampling theorem, which tells us that, under certain conditions, a signal can be perfectly reconstructed from samples at discrete points. This is the basis of all modern digital technology.

Prerequisites: MAS211 (Advanced Calculus and Linear Algebra)
No other modules have this module as a prerequisite.

## Outline syllabus

• Signals in Hilbert space.
• The Fourier Series.
• The Fourier Transform and its properties.
• Convolution, energy and bandwidth.
• Delta functions.
• Linear shift invariant (LSI) systems.
• The Shannon-Whittaker sampling theorem.

## Aims

• To develop the idea that a signal can be treated as a set of frequencies by using the Fourier transform.
• To exploit this representation to give fundamental insight into how systems act on signals.
• To demonstrate that a continuous (analog) signal can be sampled to produce a discrete (digital) signal, without any loss of information, as long as the signal contains only a finite range of frequencies.
• To convey the immense importance of these ideas to modern life.

## Learning outcomes

• understand the geometry of signals as carried by their Hilbert space (inner product space) structure;
• be able to relate physical concepts such as energy or power to Hilbert space concepts;
• understand and be able to use the frequency space (Fourier Transform) representation of signals;
• know what a linear shift invariant system is and how to model it in the time and frequency domains;
• know what a bandlimited signal is and appreciate its significance for perfect reconstruction of a continuous (analog) signal from its discrete (digital) samples.

## Teaching methods

Lectures

20 lectures, no tutorials

## Assessment

One formal 2 hour written examination. Format: 4 questions from 4.

## Full syllabus

1. Signals
Signals in Hilbert space: Definition of a signal as a member of an inner product space of functions. (Revision of ideas of norm and inner product and their significance through examples in 2-D and 3-D space. Definition of signal energy and power, relation to norm, and special signal spaces: finite energy, finite power, periodic, time-limited. Special signals: square (pa(t)) and triangular (qa(t)) pulses, step function (U(t)) and the sinc function. The Cauchy-Schwartz and triangle inequalities.

2. Fourier Analysis
The Fourier Series: Orthogonality, orthogonal bases (relation to coordinate systems) and Parseval's theorem. Crucial fact: for the set of finite power periodic signals of period T, the family of complex exponentials einσt, −∞ < n < ∞ is an orthogonal basis. Application to real and complex Fourier exponentials e series and physical interpretation including Parseval's theorem: power spectrum. The Fourier Transform and its properties: The Fourier Transform and the inverse Fourier Transform. Calculating the FT of even and odd signals: examples to include pa(t), qa(t) and U(t)exp(at). Use of FT in integration and relation of FT to Fourier series; using the FT to calculate Fourier coefficients forperiodic signals. Equivalence of operations and properties in time and frequency domains: linearity, scaling, shifting, frequency shift, symmetries, differentiation, moments, duality. Phase and amplitude spectra and their symmetries for real signals.
3. Convolution
Convolution, energy and bandwith: Convolution and its properties (linear, commutative, associative). Visualizing convolution. The convolution, product, isometry and energy (Parseval's) theorems. Band-limited signals, equivalent rectangle resolution and the time-bandwidth theorem. Physical significance of this theorem and the general concept of bandwith.
4. More Signals
Delta functions and their properties: The FT of a delta function and ensuing results on the FT of pure frequencies. The delta function as a unit for convolution and as a shift operator. The comb function and its use in sampling and in representing periodic signals.
5. LSI Systems
Linear shift invariant (LSI) systems and examples (including simple circuits): The crucial property that any exponential signal is an eigenfunction of an LSI system, which gives rise to the concept of the system transfer function (STF) and hence (through the FT) to frequency domain processing. The impluse response function (IRF) and time domain processing. Any LSI system can be described in the frequency domain by a convolution with the IRF or in the frequency domain by a convolution with the IRF or in the frequency domain by multiplication with STF. Response of an LSI system to sinusoidal inputs. Ideal filters; lowpass, highpass, and filters arising from circuits (using impedance ideas), presumming, integration, etc.
6. Sampling and digital signals
The Shannon-Whittaker sampling theorem: Sampling of a signal and the Shannon-Whittaker sampling theorem. The Nyquist frequency and the sinc function as a perfect interpolator for signals sampled faster than their Nyquist frequency. Aliasing and its physical explanation. The Poisson sum formula as a second proof of the sampling theorem and its use in interpreting the effects of sinc interpolation of undersampled signals. (Optional: deriving the sampling theorem from Hilbert space concepts and the completeness of the exponential family in the space of periodic functions).