Overview:

Continuous functions

Periodic functions; fundamental period and frequency

Complex exponential function

Linear combination of periodic functions

Energy and Power functions

Reflection, translation, scaling, stretch, compression

Unit step function

Rectangular impulse function

Signum function

Ramp function

Sampling function

Unit impulse function (Dirac delta function)

Properties of Unit impulse function

  • A Continuous Function: f(t) is said to be continuous at t0 if for every there exists  such that

| f(t) - f(t0 ) | < whenever | t - t0 | < 

  • Another definition: Let and. If = = then f is continuous at t0 .

If a function is continuous at every point in its domain, it is called a continuous function.

  • “Continuous-time function f(t) over [a,b]” --- The function f(t) need not be continuous. The domain on which it is defined is continuous.
  • If x(t) satisfies x(t) = x(t + nT) where n = 1, 2, 3, … then x(t) is periodic of period T.

Example:

(i) x(t) = A sin(t) is periodic of fundamental period 2

(ii) t = A sin(t) is periodic of period 2Here is called the frequency of the sinusoidal function.

  • Consider a collection of complex exponential functions defined by = where , k = 1, 2, 3, …., and is some real number.

Question: Is periodic? If yes, what is the period?

  • Let us assume that x(t) and y(t) are periodic with fundamental periods T1 and T2 respectively.

Question: Is z(t) = a x(t) + b y(t) periodic ( a and b are some real numbers)?

“ The linear combination of two periodic functions is periodic if the ratio of their fundamental periods is a rational number”

  • Energy and power signals:

Let, x(t) = voltage across a resistance R.

i(t) = current produced

then i(t) = x(t) / R.

The instantaneous power = R i2 ( = x2 / R) and the energy expended during the incremental time interval dt is ( x2 / R dt ).

Let us assume R = 1 ohm. Then the energy over time interval of length 2L is given by

.

Total Energy =

Average power P =

  • Consider the function x(t).

-x(t) / Reflection in t-axis
x(-t) / Reflection in x-axis
x(t) + c / Translation along x-axis by c units; positive c  up
x(t+c) / Translation along t-axis by c units; positive c  left
x(ct) / c < 1  Stretch by a factor of c along t axis;
x(ct) / c > 1  compression by a factor of c along t axis;
c x(t) / Stretch/compression by a factor of c along x-axis

Example:

(1)Sketch the graph of x(t) = sin(t)

(2)Sketch the graph of x(t) = 2 sin(/4 –t) + 1/2

Some special functions

  • Unit Step function:u(t) =

Sketch u(t-a), u(t+a) and [u(t+a) – u(t-a)]

  • Rectangular pulse function: rect(t/) =

Example: Sketch rect(t/2a) and compare this with [u(t+a) – u(t-a)]

  • Signum function: Sgn(t) =
  • Ramp function: r(t) =
  • Sampling function: Sa(t) = sin(t) / t

Sinc function: Sinc(t) = sin(t) / t

  • Unit Impulse function: (t)  “Dirac delta function”

By technical definition of a function, the Dirac delta function is not a function. (t) has following properties:

(1)(0)  infinity

(2)(t) = 0, t not equal to 0.

(3)

(4)(t) = ( - t)

Example:

Consider P1(t) =

It can be shown that P1(t) = (t)

Note that P1(t) can be rewritten as

P1(t) =

  • Any function x(t) can be approximated by infinite sum of weighted rectangular pulse functions of varying heights.

Suppose we know x(t) at points {234k

Recall, rect .

rect

x(k) rect

For infinitesimally small let us denote  by d and k by 

The right hand side reduces to (recall the Riemann Sum definition of an integral) .

Example: Evaluate

Sifting Property; Sampling Property:

Scaling Property; Derivative

  • Even and Odd functions:

x(t) is even if x(- t) = x(t) for all t.

x(t) is odd if x(-t) = - x(t) for all t.

Example 1:

x(t) = cos(t) and x(t) = t2+ 4t4 are even functions. Verify.

Example 2:

x(t) = sin(t) and x(t) = 2t+ 3t3 are odd functions. Verify.

Example 3:

x(t) = (t–2)2 is neither odd nor even. Verify.

  • Any arbitrary function x(t) can be written as sum of two function xe(t) and xo(t) where xe(t) is an even function and xo(t) is an odd function.

Let x(t) be an arbitrary function. Let us assume that there exists an even function xe(t) and an odd function xo(t) such that

x(t) = xe(t)+ xo(t)

then x(-t) = xe(-t)+ xo(-t) = xe(t)- xo(t)

By solving these two equations weget

xe(t)= 1/2[x(t) + x(-t)] and xo(t) = 1/2 [x(t) – x(-t)]

Exercise: Show that x(t) = (t–1)2+ sin (t) is neither even nor odd. Find an even functionxe(t)and an odd functionxo(t) such that

x(t) = xe(t)+ xo(t)

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