SIGNALS AND TRANSFORMS
Course Objectives:
- To provide the basic understanding of signals and systems
- To provide the knowledge of time- domain and frequency- domain representation of continuous-time & discrete-time periodic & a periodic signals
- Signals and systems
(7 hours)
- Signal definition, continuous time signal, discrete time signal
- Basic signal types: energy signal and power signal, periodic and aperiodic signal
- Transformation of independent variable
- Unit impulse and unit step functions
- System definition, continuous time system, discrete time system
- Basic system properties: system with and memory, invertibility, causality, stability, time invariance, linearity
- Impulse Reponse and Convolution
(6 hours)
- Linear time invariant (LTI) systems and their properties
- Continuous time LTI systems: Representation of continuous time signals in terms of impulses, Unit impulse response and convolution integral
- Discrete time LTI systems: Representation of discrete time signals in terms of impulses, Unit impulse response and Convolution sum
- Fourier Series
(9 hours)
- Fourier series representation of continuous time periodic signals
- Properties of continuous time Fourier series (linearity, time shift, time reversal. Time scaling, multiplication, conjugation, convolution), Parseval’s relation
- Fourier series representation of discrete time periodic signals
- Properties of discrete time Fourier series, Parseval’s relation
- Fourier Transforms
(9 hours)
- Fourier transform representation of continuous time aperiodic signal
- Properties of continuous time Fourier tranform (linearity, time shift, time scaling, frequency scaling, conjugation, duality), Parseval’s relation, Energy spectral density and Power spectral density
- Discrete time Fourier transform, Properties of discrete time Fourier transform
- Introduction to discrete time Fourier (DFT) and fast Fourier transform (FFT)
- Sampling Theory
(5 hours)
- Representation of continuous time signal by its samples: Sampling
- Spectral properties of sampled signal
- Reconstruction of signal from its samples
- Ideal sampling and practical considerations in sampling
- Effect of under sampling: Aliasing
- Z-transform and Difference Equations
(9 hours)
- Z-transform, convergence and region of convergence.
- Properties of Z-transform (linearity, time shift, time reversal, multiplication by exponential sequence, differentiation, convolution, multiplication)
- Z-Transform pairs
- Inverse Z-transform by long division and partial fraction expansion
- Discrete time systems described by difference equations
- Solution of difference equations
Practical:
- Signals and transform
- Impulse Response and Convolution
- Fourier Series
- Fourier Transform
- Sampling Theory
- Z-transform and Difference Equations
Reference:
- V. Oppenhiem, A.S. Willsky, S.H. Nawab, “Signals and Systems”, Prentice Hall
- G. Proakis, D.G. Manolakis, “Digital Signal Processing”, Prentice Hall
Assessments: Mean of three
Evaluation Scheme:
Marks distribution for all the chapters in the syllabus in shown in the table below.
Unit |
Hours |
Marks distribution* |
1,5.1-5.3 |
7+2 |
16 |
2,5.4,5.6 |
6+3 |
16 |
3 |
9 |
16 |
4 |
9 |
16 |
6 |
9 |
16 |
Total |
45 |
80 |
*There may be minor variation in marks distribution.
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