### Signals and Systems Syllabus

- Signals
- Mathematical description and pictorial representation of commonly used continuous-time signals and discrete time signals such as rectangular signal, unit step, dirac-delta, ramp, sinusoidal, complex exponential signals, sinc
- Even and odd signals, periodic signals
- Transformations of the independent variable – shift in time, scaling of the time axis
- Signal energy, power, auto-correlation, cross correlation, sifting property of the impulse

- Basic properties of systems
- Systems with and without memory, linearity, invertibility, causality, stability, time invariance.

- Linear Time – Invariant Systems
- Impulse response of a system
- Convolution in discrete-time and continuous-time
- Properties of LTI systems – commutative property, distributive property, associative property, invertibility, causality, stability
- LTI systems described by differential (or, difference) equations
- Block diagram representation of systems represented by differential (or, difference) equations
- Eigen functions of LTI systems

- Fourier series representation of periodic signals
- Determination of trigonometric and complex exponential Fourier series for continuous time and discrete time periodic signals
- Convergence of the Fourier series
- Properties of the FS – linearity, shifting in time, scaling of the time axis, multiplication, conjugation, conjugate symmetry, Parseval’s identity (See also section of properties of the Fourier Transform)

- Continuous-time and discrete-time Fourier transform
- Development of the Fourier transform of an aperiodic signal
- Dirichlet conditions, convergence of the Fourier transform
- Computing the Fourier transform from the definition
- Memorize Fourier transform of basic signals such as rectangular signal, sinc, delta, exponential signal
- Properties of the Fourier transform – linearity, time shift, frequency shift, scaling of the time axis and frequency axis, conjugation and symmetry, time reversal, differentiation and integration, duality, Parseval’s relation. Be conversant in using the properties of Fourier transforms to compute the FT of signals that can be obtained from simpler signals through a series of the above operations.
- Convolution and multiplication property
- Inverse Fourier transform – be able to compute this from definition as well as from looking up the transform for elementary signals. Be able to use partial fraction expansions to compute the Inverse Fourier transform.
- Magnitude and phase representation of the Fourier transform and frequency response of LTI systems

- Applications of the Frequency domain analysis of signals and
systems
- Filtering – Frequency response and impulse response of ideal filters, first order and second order approximations to filters.
- Sampling – Nyquist theorem, effects of aliasing, ideal reconstruction of the signal from its samples
- Modulation – Amplitude modulation, Hilbert transform, DSB and SSB carrier modulation

- Laplace Transforms
- Definition, region of convergence, inverse Laplace transform
- Pole-Zero plot
- Properties of the Laplace transform - – linearity, time shift, frequency shift, scaling of the time axis and frequency axis, conjugation and symmetry, time reversal, differentiation and integration, duality, Parseval’s relation, initial and final value theorems
- Solving differential equations using Laplace transform

- Z-transforms
- Definition of direct z-transform, region of convergence (ROC), inverse z-transform using partial fraction expansion
- Pole-zero plot
- Properties of Z-transform -- linearity, time shift, z-scaling, time reversal, conjugation, z-differentiation, convolution, stability and its relation to causality and ROC
- Transfer function of discrete-time systems and analysis of systems described by constant coefficient difference equations