Signals and systems This subject deals with mathematical methods used to describe signals and to analyze and synthesize systems. Signals are variables that carry informationSignals are variables that carry information Systems process input signals to produce output signals.Systems process input signals to produce output signals.

Examples of signals Electrical signals --- voltages and currents in a circuit Acoustic signals --- audio or speech signals (analog or digital) Video signals --- intensity variations in an image (e.g. a CAT scan) Biological signals --- sequence of bases in a gene M Noise is an unwanted signals

Examples of signals The variables can also be spatial Eg. Cervical MRI In this example, the signal is the intensity as a function of the spatial variables x and y.

Examples of signals

Bounded and Unbounded Signals Whether the output signal of a system is bounded or unbounded determines the stability of the system. Eg. Demo, the inverted pendulum, unstable but can be stabilized using a feedback control system.

Periodic and aperiodic signals Periodic signals have the property that x(t + T) = x(t) for all t. The smallest value of T that satisfies the definition is called the period. Shown below are an aperiodic signal (left) and a periodic signal (right).

Unit impulse — definition The unit impulse δ(t), aka the Dirac delta function, is not a function in the ordinary sense. It is defined by the integral relation. and is called a generalized function. The unit impulse is not defined in terms of its values, but is defined by how it acts inside an integral when multiplied by a smooth function f(t).

Unit impulse — narrow pulse approximation To obtain an intuitive feeling for the unit impulse, it is often helpful to imagine a set of rectangular pulses where each pulse has width е and height 1/е so that its area is 1. The unit impulse is the quintessential tall and narrow pulse!

Unit impulse — the shape does not matter A triangular pulse approximation is just as good. As far as our definition is concerned both the rectangular and triangular pulse are equally good approximations. Both act as impulses. There is nothing special about the rectangular pulse approximation to the unit impulse.

Real and complex signals The signals can Real, Imaginary or Complex. CT signal x(t) = e st where is s is a complex number, DT signal x[n] = z n where z is a complex number.Question Why do we deal with complex signals ?

Real and complex signals

Even and odd signals Even signals x e (t) and odd signals x o (t) are defined as: x e (t) = x e (-t) and x o (t) = -x o (-t)

Complex Signals

Problem 1-1 Interpret and sketch the generalized function x(t) where x(t) = e t/4 δ(t+4).

Problem 1-1 Solution To determine the meaning of x(t) we place it in an integral:- Let τ = t+4 so that From the definition of the unit impulse, the integral equals e.1. Therefore,

Problem 1-1 Solution

Unit step

Unit impulse as the derivative of the unit step As an example of the method for dealing with generalized functions consider the generalized function:- Since u(t) is discontinuous, its derivative does not exist as an ordinary function, but it does as a generalized function. To see what x(t) means, put it in an integral with a smooth testing function and apply the usual integration-by-parts theorem to obtain

Unit impulse as the derivative of the unit step, cont’d The result is that:- which, from the definition of the unit impulse, implies that That is, the unit impulse is the derivative of the unit step in a generalized function sense.