Digital Signal Processing Course

Lesson 1 • System Properties and Classification

Linearity, time-invariance, causality, stability, and memory defined and tested. Understanding these properties determines which analysis tools apply.

Lesson 2 • Elementary Signal Models

Unit impulse, unit step, ramp, sinusoidal, and exponential signals in both domains. These models serve as building blocks for system analysis.

Lesson 3 • Basic Signal Operations

Time shifting, scaling, reversal, and amplitude transformations applied to signals. Provides manipulation tools needed for convolution and transform analysis.

Lesson 4 • Linear Time-Invariant System Basics

LTI systems characterised by impulse response and convolution integral/sum. Connects system properties to the convolution operation introduced next.

Lesson 1 • Convolution Sum Computation

Analytical and graphical methods for computing the discrete convolution sum. Mastery here is prerequisite for frequency-domain analysis.

Lesson 2 • Discrete-Time Signal Representation

Sequences, index notation, and graphical representation of discrete-time signals. Reinforces continuous-time concepts in the discrete domain.

Lesson 3 • Stability and Causality via Convolution

BIBO stability and causality conditions expressed in terms of the impulse response. Connects abstract system properties to computable criteria.

Lesson 4 • Discrete-Time LTI Systems

Impulse response, difference equations, and input-output relationships for discrete LTI systems. Bridges system theory to practical digital filter structures.

Lesson 1 • Continuous-Time Fourier Series

Synthesis and analysis equations, coefficient computation, and convergence behaviour. Provides the mathematical core of frequency-domain representation.

Lesson 2 • Discrete-Time Fourier Series

Fourier series for discrete periodic sequences, including periodicity of the spectrum. Bridges continuous Fourier series to the DFT introduced later.

Lesson 3 • Periodic Signals and Harmonics

Fundamental period, fundamental frequency, and harmonic relationships in periodic signals. Sets the conceptual stage for Fourier decomposition.

Lesson 4 • Fourier Series Spectrum

Magnitude and phase spectra plotted from Fourier coefficients for real signals. Develops spectral visualisation skills used in all subsequent chapters.

Lesson 1 • Discrete-Time Fourier Transform

DTFT definition, periodicity, and key transform pairs for discrete sequences. Extends spectral analysis to discrete-time signals processed by digital systems.

Lesson 2 • DTFT Properties and Applications

Linearity, time shift, frequency shift, convolution, and Parseval properties of the DTFT. Enables efficient analysis of discrete LTI systems in the frequency domain.

Lesson 3 • CTFT Properties and Theorems

Linearity, time shift, frequency shift, scaling, duality, convolution, and Parseval theorems. These properties enable efficient transform computation without direct integration.

Lesson 4 • Spectral Interpretation and Filtering

Magnitude and phase response, ideal filter characteristics, and spectral shaping concepts. Connects transform theory to practical filter design goals.

Lesson 1 • Practical Sampling Systems

Sample-and-hold circuits, quantisation effects, and analogue-to-digital converter architecture. Connects ideal theory to real hardware constraints in data acquisition.

Lesson 2 • Aliasing and Anti-Aliasing Filters

Spectral overlap from undersampling and the role of analogue lowpass pre-filters. Practical design of anti-aliasing filters to prevent irreversible distortion.

Lesson 3 • Multirate Sampling Concepts

Downsampling, upsampling, and their spectral effects as a foundation for multirate systems. Prepares students for polyphase filter banks covered in advanced chapters.

Lesson 4 • Reconstruction and Interpolation

Ideal sinc interpolation, zero-order hold, and linear interpolation for signal reconstruction. Evaluates reconstruction quality and distortion introduced by practical methods.

Lesson 1 • Inverse Z-Transform Methods

Partial fraction expansion, power series expansion, and contour integration for inversion. Selecting the correct method depends on ROC and sequence type.

Lesson 2 • System Function and Difference Equations

Transfer function H(z) derived from difference equations and block diagram structures. Unifies time-domain and Z-domain descriptions of discrete systems.

Lesson 3 • Z-Transform Properties

Linearity, time shift, scaling, convolution, differentiation, and initial/final value theorems. Properties reduce complex transform computations to algebraic manipulations.

Lesson 4 • Z-Transform Definition and ROC

Bilateral and unilateral Z-transform definitions, region of convergence, and common pairs. The ROC determines causality and stability of the associated system.

Lesson 1 • DFT Applications and Limitations

Power spectrum estimation, frequency detection, and picket-fence effect in DFT analysis. Recognising DFT limitations prevents misinterpretation of computed spectra.

Lesson 2 • Spectral Analysis with the DFT

Zero-padding, frequency resolution, spectral leakage, and windowing effects on DFT spectra. Proper use of these techniques is essential for accurate spectral estimation.

Lesson 3 • Fast Fourier Transform Algorithms

Decimation-in-time and decimation-in-frequency FFT algorithms and their computational savings. Reduces DFT complexity from O(N²) to O(N log N) for practical implementation.

Lesson 4 • Circular Convolution and Linear Convolution

Circular convolution via DFT multiplication and its relationship to linear convolution. Overlap-add and overlap-save methods enable efficient long-sequence filtering.

Lesson 1 • Filter Specifications and Types

Passband, stopband, transition band, ripple, and attenuation specifications for digital filters. Translating application requirements into quantitative filter specs is the first design step.

Lesson 2 • IIR Filter Design Methods

Bilinear transform and impulse invariance methods mapping analogue prototypes to digital IIR filters. Butterworth, Chebyshev, and elliptic prototypes provide different ripple trade-offs.

Lesson 3 • FIR Filter Design Methods

Window method, frequency sampling, and Parks-McClellan equiripple design for FIR filters. FIR filters guarantee linear phase and unconditional stability.

Lesson 4 • Finite Wordlength Effects

Coefficient quantisation, roundoff noise, overflow, and limit cycles in fixed-point implementations. Mitigating these effects is critical for reliable embedded DSP systems.