Signals and SystemsSignals and Systems

Lecture #4Lecture #4

•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

Continuous Time SignalsContinuous Time Signals

Construction of the Unit-impulse function Construction of the Unit-impulse function ((tt))

One of the simplest way –– rectangularOne of the simplest way –– rectangular

pulse, taking the limit pulse, taking the limit → 0 → 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 pulsesare short enough –– much shorter than the characteristic time scale of the system.

Response of a CT LTI Response of a CT LTI SystemSystem

Now suppose the system is LTI, and define the unitimpulse response h(t):

From Time-Invariance:

From Linearity:

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

DistributivityDistributivity

Properties of the CTFTProperties of the CTFT• Properties — symmetryProperties — symmetry

We start with the definition of the Fourier We start with the definition of the Fourier transform of a real time function transform of a real time function xx((tt) and ) and expand both terms in the integrand inexpand both terms in the integrand in

terms of odd and even components.terms of odd and even components.

The even components of the integrand contribute zero to the integral. Hence, we obtain

Unit impulse — what do we need it Unit impulse — what do we need it for?for?

The unit impulse is a valuable The unit impulse is a valuable idealization and is used widely in idealization and is used widely in science and engineering. science and engineering. Impulses in time are useful Impulses in time are useful idealizations.idealizations.

• Impulse of current in time Impulse of current in time delivers a unit charge delivers a unit charge instantaneously to a network.instantaneously to a network.

• Impulse of force in time delivers Impulse of force in time delivers an instantaneous momentum to a an instantaneous momentum to a mechanical system.mechanical system.

• Impulse of mass density in space Impulse of mass density in space represents a point mass.represents a point mass.

what do we need it for?.....cont.what do we need it for?.....cont.• Impulse of charge density in Impulse of charge density in

space represents a point charge.space represents a point charge.• Impulse of light intensity in Impulse of light intensity in

space represents a point of light.space represents a point of light.

We can imagine impulses in space We can imagine impulses in space and time.and time.

• Impulse of light intensity in Impulse of light intensity in space and time represents a brief space and time represents a brief flash of light at a point in space.flash of light at a point in space.

Unit stepUnit step

Integration of Integration of the unit impulse the unit impulse yields the unit yields the unit step function:step function:

which is defined as

Unit impulse as the derivative of the Unit impulse as the derivative of the unit stepunit step

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

Successive integration of the unit Successive integration of the unit impulseimpulse

ConclusionsConclusions• We are awash in a sea of signals.We are awash in a sea of signals.• Signal categories — identity of Signal categories — identity of

independent variable, independent variable, dimensionality, CT & DT, real & dimensionality, CT & DT, real & complex, periodic & aperiodic, complex, periodic & aperiodic, causal & anti-causal, bounded & causal & anti-causal, bounded & unbounded, even & odd, etc.unbounded, even & odd, etc.

• Building block signals — eternal Building block signals — eternal complex exponentials and impulse complex exponentials and impulse functions — are a rich class of functions — are a rich class of signals that can be superimposed to signals that can be superimposed to represent virtually any signal of represent virtually any signal of physical interest.physical interest.