Digital Signal Processing – Chapter 2 Study Notes
Based on the topics discussed: discrete-time signals, systems, convolution, and difference
equations.
1. Discrete-Time Signals
A discrete-time signal is a sequence of numbers defined at discrete instants of time. It is usually
represented as x(n), where n is an integer index. Common operations include time shifting, scaling,
folding (time reversal), and addition of signals.
2. Discrete-Time Systems
A system transforms an input signal x(n) into an output signal y(n). Important properties include
linearity, time invariance, causality, and stability.
3. LTI Systems
Linear Time-Invariant (LTI) systems are completely characterized by their impulse response h(n).
The output can be determined using convolution.
4. Convolution
For discrete-time signals, convolution is defined as: y(n) = Σ x(k) h(n − k). The graphical method
consists of folding one signal, shifting it, multiplying overlapping samples, and summing the
products.
5. Difference Equations
Many discrete-time systems are described by difference equations. For example: y(n) − 0.5y(n−1) =
x(n). These equations relate present and past input/output samples.
6. Correlation
Correlation measures the similarity between two signals. It is widely used in detection, pattern
recognition, and communications.
Useful Tips
- Identify whether a signal is finite or infinite. - Check linearity and time invariance separately. - For
convolution, determine the overlap interval first. - Verify causality and stability when analyzing
systems. - Practice graphical convolution because it appears frequently in exercises.