Signals and Systems

1. Signals and Systems • Representation of CT Signals in terms of shifted unit impulses • Introduction of the unit impulse (t) • Convolution integral representation of CT LTI systems • Properties and Examples Lecture #4

2. Continuous Time Signals

3. Construction of the Unit-impulse function (t) One of the simplest way –– rectangular pulse, taking the limit  → 0. But this is by no means the only way. One can construct a (t) function out of many other functions, e.g. Gaussian pulses, triangular pulses, sinc functions, etc., as long as the pulses are short enough –– much shorter than the characteristic time scale of the system.

4. Response of a CT LTI System Now suppose the system is LTI, and define the unit impulse response h(t): From Time-Invariance: From Linearity:

5. Convolution A convolution is an integral that expresses the amount of overlap of one function g as it is shifted over another function f. It therefore "blends" one function with another. For example, in synthesis imaging, the measured dirty map is a convolution of the "true" CLEAN map with the dirty beam (the Fourier transform of the sampling distribution). Abstractly, a convolution is defined as a product of functions f and g that are objects in the algebra of Schwartz functions in Convolution of two functions f and g over a finite range is given by:

7. Distributivity

8. Properties of the CTFT • Properties — symmetry We start with the definition of the Fourier transform of a real time function x(t) and expand both terms in the integrand in terms of odd and even components. The even components of the integrand contribute zero to the integral. Hence, we obtain

9. Unit impulse — what do we need it for? The unit impulse is a valuable idealization and is used widely in science and engineering. Impulses in time are useful idealizations. • Impulse of current in time delivers a unit charge instantaneously to a network. • Impulse of force in time delivers an instantaneous momentum to a mechanical system. • Impulse of mass density in space represents a point mass.

10. what do we need it for?.....cont. • Impulse of charge density in space represents a point charge. • Impulse of light intensity in space represents a point of light. We can imagine impulses in space and time. • Impulse of light intensity in space and time represents a brief flash of light at a point in space.

11. Unit step Integration of the unit impulse yields the unit step function: which is defined as

12. Unit impulse as the derivative of the unit step

13. Unit impulse as the derivative of the unit step, cont’d

14. Successive integration of the unit impulse

17. Conclusions • We are awash in a sea of signals. • Signal categories — identity of independent variable, dimensionality, CT & DT, real & complex, periodic & aperiodic, causal & anti-causal, bounded & unbounded, even & odd, etc. • Building block signals — eternal complex exponentials and impulse functions — are a rich class of signals that can be superimposed to represent virtually any signal of physical interest.