UNIT II Analysis of Continuous Time signal

Introduction Decompose a periodic input signal into primitive periodic components. A periodic sequence T 2T 3T t f(t)

T is a period of all the above signals Synthesis T is a period of all the above signals Even Part Odd Part DC Part Let 0=2/T.

Decomposition

Example (Square Wave)  2 3 4 5 - -2 -3 -4 -5 -6 f(t) 1

Example (Square Wave)

T is a period of all the above signals Harmonics T is a period of all the above signals Even Part Odd Part DC Part

Harmonics Define , called the fundamental angular frequency. Define , called the n-th harmonic of the periodic function.

Harmonics

Amplitudes and Phase Angles harmonic amplitude phase angle

Complex Form of Fourier Series

Complex Exponentials

Complex Form of the Fourier Series

Complex Form of the Fourier Series

Complex Form of the Fourier Series

Complex Form of the Fourier Series If f(t) is real,

Example t f(t) A

Example A/5 40 80 120 -40 -120 -80 50 100 150 -50 -100 -120 -80 A/5 50 100 150 -50 -100 -150

Example A/10 40 80 120 -40 -120 -80 100 200 300 -100 -200 -120 -80 A/10 100 200 300 -100 -200 -300

Example t f(t) A

Waveform Symmetry Even Functions Odd Functions

Decomposition Any function f(t) can be expressed as the sum of an even function fe(t) and an odd function fo(t). Even Part Odd Part

Example Even Part Odd Part

Half-Wave Symmetry and T T/2 T/2

Quarter-Wave Symmetry Even Quarter-Wave Symmetry T T/2 T/2 Odd Quarter-Wave Symmetry T T/2 T/2

Hidden Symmetry The following is a asymmetry periodic function: Adding a constant to get symmetry property. A/2 A/2 T T

Fourier Coefficients of Symmetrical Waveforms The use of symmetry properties simplifies the calculation of Fourier coefficients. Even Functions Odd Functions Half-Wave Even Quarter-Wave Odd Quarter-Wave Hidden

Fourier Coefficients of Even Functions

Fourier Coefficients of Even Functions

Fourier Coefficients for Half-Wave Symmetry and T T/2 T/2 The Fourier series contains only odd harmonics.

Fourier Coefficients for Half-Wave Symmetry and

Fourier Coefficients for Even Quarter-Wave Symmetry

Fourier Coefficients for Odd Quarter-Wave Symmetry

Example Even Quarter-Wave Symmetry T T/2 T/2 1 1 T T/4 T/4

Example Even Quarter-Wave Symmetry T T/2 T/2 1 1 T T/4 T/4

Example Odd Quarter-Wave Symmetry T T/2 T/2 1 1 T T/4 T/4

Example Odd Quarter-Wave Symmetry T T/2 T/2 1 1 T T/4 T/4

Dirichlet Conditions A periodic signal x(t), has a Fourier series if it satisfies the following conditions: x(t) is absolutely integrable over any period, namely x(t) has only a finite number of maxima and minima over any period x(t) has only a finite number of discontinuities over any period

Fourier Transform We have seen that periodic signals can be represented with the Fourier series Can aperiodic signals be analyzed in terms of frequency components? Yes, and the Fourier transform provides the tool for this analysis The major difference w.r.t. the line spectra of periodic signals is that the spectra of aperiodic signals are defined for all real values of the frequency variable not just for a discrete set of values

The Fourier Transform in the General Case Given a signal x(t), its Fourier transform is defined as A signal x(t) is said to have a Fourier transform in the ordinary sense if the above integral converges

The Fourier Transform in the General Case – Cont’d The integral does converge if the signal x(t) is “well-behaved” and x(t) is absolutely integrable, namely, Note: well behaved means that the signal has a finite number of discontinuities, maxima, and minima within any finite time interval

Rectangular Form of the Fourier Transform Consider Since in general is a complex function, by using Euler’s formula

Polar Form of the Fourier Transform can be expressed in a polar form as where

Fourier Transform of Real-Valued Signals If x(t) is real-valued, it is Moreover whence Hermitian symmetry

Example: Fourier Transform of the Rectangular Pulse Consider the even signal It is

Example: Fourier Transform of the Rectangular Pulse – Cont’d

Example: Fourier Transform of the Rectangular Pulse – Cont’d amplitude spectrum phase spectrum

Bandlimited Signals A signal x(t) is said to be bandlimited if its Fourier transform is zero for all where B is some positive number, called the bandwidth of the signal It turns out that any bandlimited signal must have an infinite duration in time, i.e., bandlimited signals cannot be time limited

Bandlimited Signals – Cont’d If a signal x(t) is not bandlimited, it is said to have infinite bandwidth or an infinite spectrum Time-limited signals cannot be bandlimited and thus all time-limited signals have infinite bandwidth However, for any well-behaved signal x(t) it can be proven that whence it can be assumed that B being a convenient large number

Inverse Fourier Transform Given a signal x(t) with Fourier transform , x(t) can be recomputed from by applying the inverse Fourier transform given by Transform pair

Properties of the Fourier Transform Linearity: Left or Right Shift in Time: Time Scaling:

Properties of the Fourier Transform Time Reversal: Multiplication by a Power of t: Multiplication by a Complex Exponential:

Properties of the Fourier Transform Multiplication by a Sinusoid (Modulation): Differentiation in the Time Domain:

Properties of the Fourier Transform Integration in the Time Domain: Convolution in the Time Domain: Multiplication in the Time Domain:

Properties of the Fourier Transform Parseval’s Theorem: Duality: if

Properties of the Fourier Transform - Summary

Example: Linearity

Example: Time Shift

Example: Time Scaling time compression frequency expansion time expansion frequency compression

Example: Multiplication in Time

Example: Multiplication by a Sinusoid sinusoidal burst

Example: Multiplication by a Sinusoid – Cont’d

Example: Multiplication in Time

Example: Multiplication by a Sinusoid sinusoidal burst

Example: Multiplication by a Sinusoid – Cont’d

Example: Integration in the Time Domain

Example: Integration in the Time Domain – Cont’d The Fourier transform of x(t) can be easily found to be Now, by using the integration property, it is

Example: Integration in the Time Domain – Cont’d

Generalized Fourier Transform Fourier transform of Applying the duality property generalized Fourier transform of the constant signal

Generalized Fourier Transform of Sinusoidal Signals

Fourier Transform of Periodic Signals Let x(t) be a periodic signal with period T; as such, it can be represented with its Fourier transform Since , it is

Fourier Transform of the Unit-Step Function Since using the integration property, it is

Common Fourier Transform Pairs

Laplace Transform

Why use Laplace Transforms? Find solution to differential equation using algebra Relationship to Fourier Transform allows easy way to characterize systems No need for convolution of input and differential equation solution Useful with multiple processes in system

How to use Laplace Find differential equations that describe system Obtain Laplace transform Perform algebra to solve for output or variable of interest Apply inverse transform to find solution

How to use Laplace Find differential equations that describe system Obtain Laplace transform Perform algebra to solve for output or variable of interest Apply inverse transform to find solution

What are Laplace transforms? t is real, s is complex! Inverse requires complex analysis to solve Note “transform”: f(t)  F(s), where t is integrated and s is variable Conversely F(s)  f(t), t is variable and s is integrated Assumes f(t) = 0 for all t < 0

Evaluating F(s) = L{f(t)} Hard Way – do the integral let let let

Evaluating F(s)=L{f(t)}- Hard Way remember let Substituting, we get: let It only gets worse…

Table of selected Laplace Transforms

More transforms

Note on step functions in Laplace Unit step function definition: Used in conjunction with f(t)  f(t)u(t) because of Laplace integral limits:

Properties of Laplace Transforms Linearity Scaling in time Time shift “frequency” or s-plane shift Multiplication by tn Integration Differentiation

Properties of Laplace Transforms Linearity Scaling in time Time shift “frequency” or s-plane shift Multiplication by tn Integration Differentiation

Properties: Linearity Example : Proof :

Properties: Scaling in Time Example : Proof : let

Properties: Time Shift Example : Proof : let

Properties: S-plane (frequency) shift Example : Proof :

Properties: Multiplication by tn Example : Proof :

The “D” Operator Differentiation shorthand Integration shorthand if if then then

Properties: Integrals Proof : Example : let If t=0, g(t)=0 for so slower than

Properties: Derivatives (this is the big one) Example : Proof : let

The Inverse Laplace Transform

Inverse Laplace Transforms Background: To find the inverse Laplace transform we use transform pairs along with partial fraction expansion: F(s) can be written as; Where P(s) & Q(s) are polynomials in the Laplace variable, s. We assume the order of Q(s) P(s), in order to be in proper form. If F(s) is not in proper form we use long division and divide Q(s) into P(s) until we get a remaining ratio of polynomials that are in proper form.

Inverse Laplace Transforms Background: There are three cases to consider in doing the partial fraction expansion of F(s). Case 1: F(s) has all non repeated simple roots. Case 2: F(s) has complex poles: (expanded) Case 3: F(s) has repeated poles. (expanded)

Inverse Laplace Transforms Case 1: Illustration: Given: Find A1, A2, A3 from Heavyside

Inverse Laplace Transforms Case 3: Repeated roots. When we have repeated roots we find the coefficients of the terms as follows:

Inverse Laplace Transforms Case 3: Repeated roots. Example ? ? ?

Inverse Laplace Transforms Complex Roots: An Example. For the given F(s) find f(t)

Inverse Laplace Transforms Complex Roots: An Example. (continued) We then have; Recalling the form of the inverse for complex roots;