DIGITAL SIGNAL PROCESSING BASICS
A Four-Day Intensive Course
Digital Signal Processing (“DSP”) lies at the heart of most modern instrumentation and communication systems. Fields as diverse as domestic audio systems, radar and sonar systems and high-speed digital communications rely heavily on DSP concepts and on their hardware embodiment in the form of specialist microprocessors, Field-Programmable Gate Arrays and custom integrated circuits. DSP is a mathematically rich discipline that draws on elements of numerical analysis, matrix theory, transform theory and electronic systems. Students of DSP must employ mixtures of hand derivations, hand calculations and powerful computer tools when navigating through DSP analysis and synthesis tasks. This course gives an introductory, yet fairly comprehensive, overview of these issues and how they interact when gaining understanding of DSP systems and engaging in their design. Several applications receive attention on the final day of the course, furnishing an opportunity to exercise insights facilitated by earlier study of theoretical aspects of DSP.
2. Structure of the Four Days of Study
Each day of study is built around three core lecture sessions which are peppered with frequent hands-on computer-based confirmation and extension activities. These are followed by small design tasks aimed at consolidation of each session’s material. At least one Laboratory Investigation is undertaken daily to promote both individual reflection and small-group discussion of design issues. In total there is structured study time of about 8 hours per day.
(a) DAY 1: Discrete-Time Signals and Spectra
Review of continuous-time signals and their Fourier transforms: pulses, impulses, sincs, sinusoids, complex exponentials and tonebursts; shifting in time and frequency; duality; how transfer functions for continuous-time systems modify spectra; the Sampling Theorem; aliasing; distinction between discrete-time and digital signals; quantization; A/D conversion; first taste of the effects of poor anti-aliasing and reconstruction filtering.
Direct numerical generation of real and complex discrete-time signals; energy signals; discrete-time sequence counterparts of popular continuous-time signals; cautions on “digitizing”; important summation closed forms; from the Fourier transform to the Discrete-Time Fourier Transform (DTFT); sampling the DTFT to get the DFT.
Power signals and convergence difficulties for the DTFT; DTFT properties; interplay of record length, spectral resolution and sampling frequency; the power of the FFT; care in spectral sample ordering and padding for coarsegrain resolution control; examining chirps, multi-tone bursts and step-chirps; manipulations of DFT data; NDFT for probing spectra off the DFT grid; improving spectral feature-viewing by signal windowing and spectral averaging; finegrain spectrum zooming through use of the Chirp Transform Algorithm.
(b) DAY 2: FIR Digital Filters, Convolution and Adaptive FIR Filtering
Modification of signals and their spectra by both Linear Time-Invariant (LTIV) and non-LTIV system elements; why both categories are vital in practical situations; difference equations for filters; assembling and exercising an FIR structure; the Two-Point Moving Averager, its Fourier transfer function and impulse response sequence; another simple FIR: the Differencer; cascaded and paralleled filter combinations; discrete convolution; noting the duration-spreading of an FIR output signal; circular convolution from DFT products.
Variety of digital filtering roles; distortionless transmission and the opportunity for FIR phase linearity; Smagnitude defined and sought; impulse response symmetry and attractions of FIR implementations; Frequency-Sampling FIR design; quality metrics for a digital filter; Impulse-Invariant FIR design; even-length and odd-length filter distinctions; modifying designs by windowing; SNR for noisy sinusoids; quantifying SNR improvement with a digital filter.
The adaptive linear combiner; the LMS algorithm; structure of the LMS adaptive FIR filter; stylized problems suitable for configuring LMS adaptive solutions; step size selection; coefficient initialization effects; learning curves; practice with system identification and equalization of varying channel effects.
(c) DAY 3: Z-Transforms, Pole-Zero Patterns and IIR Digital Filters
The z-transform of a sequence; use for expressing difference equations as a product of an input transform and a z-domain transfer function; z-transform properties; casting cascaded and paralleled system combinations as z-domain commodities; z-plane poles and zeros and their significance for analysis and design; pole-zero patterns, understanding PZP geometry and PZP usage as signatures for filter behaviour; dangers of feedback in system structures; dictates for BIBO stability; inverse nature of Accumulators and Differencers; general closed-loop feedback z-expressions; spectral compression by zero injection; resonators and comb filters.
Group delay and its significance for signals and filters; minimum-phase, linear-phase and maximum-phase filter conditions contrasted; “half-distortion” attributes of linear-phase FIR and allpass IIR filters; observing severity of distortion damage and goals for signal restoration; cascade compensation; inverse and whitening filters; influence of phase condition on invertibility; design of allpass filters to perform phase equalization; quasi-inverse FIR filters; experiments with digital reverberation and de-reverberation.
FIR vs IIR selection tradeoffs; response transients; four IIR classics: Butterworth, Chebyshev I and II, Elliptic – their appeal and weaknesses; ease of arbitrary IIR Brandenstein-Unbehauen design; finite wordlength degradations for digital filters; perils of coefficient quantization; various measures of bandwidth for digital signals and filters.
(d) DAY 4: DSP Instrumentation and Communication Tasks
Measuring power with meters employing fixed and variable filter coefficients; importance of pulse and toneburst detection; raw voltage and energy detection schemes and SNR measures; time autocorrelation; FIR matched filters operating in white noise; pulse data communication, SNRs and error probabilities; eye diagrams and Root-Nyquist filter usage; narrowband interference removal by FIR comb and IIR notch filters; adaptive notch filtering.
Sophisticated signals having high time-bandwidth products; pulse compression in radar/sonar reception applications; use of chirps and codes for compression; revisiting A/D conversion: untangling and quantifying deficiencies in anti-aliasing and reconstruction filters, and digitally compensating to expand usable bandwidth.
Achieving fractional-sample delay in DSP; multirate DSP concepts; zero-insertion interpolation and spectral compression; FIR interpolation filters; decimation and spectral expansion; FIR decimation filters; sample-rate conversion.
3. Makeup of Course Notes
Course notes are PowerPoint slide images, with consistent notation and careful referencing of the relevant DSP literature. These bound notes are self-contained; however, students wishing to gain an alternative perspective will be encouraged to access the course reference texts: Mitra, S.K., Digital Signal Processing- A Computer-Based Approach, Third Edition, New York: McGraw-Hill, 2005, and Ambardar, A., Digital Signal Processing: A Modern Introduction, Nelson, 2007. Students receive a DVD containing all course exercises and videoclips of indicative solution approaches.
4. Computation Support
This course has an exceptionally strong hands-on flavour. All student work is reliant on the ready availability of the DSP Creations’ toolset (Slifer, Sketch-a-Filt and DSP_Speedster) which are built around the MATLAB/Simulink environment from The MathWorks Ltd. Learning motivation is sharpened by a rich blend of user-friendly graphics display and interactive instrument control features and numerous audio illustrations of the issues in digital filter design. Students leave the course with their own personal copies of Slifer Lite and Sketch-a-Filt Lite for fuller absorption of, and later reflection upon, their course experience.
5. Course Pre-requisites
Engineers, mathematicians, physicists and other technical personnel with a ready grasp of mathematical analysis (including integral calculus and fundamentals of complex variables) are the intended participants. Familiarity with filtering terminology, DSP concepts, or basics of MATLAB - while not required - would be an added advantage.