Course Outline (Tentative) Fundamental Concepts of Signals and Systems Signals Systems Linear Time-Invariant (LTI) Systems Convolution integral and sum Properties of LTI Systems … Fourier Series Response to complex exponentials Harmonically related complex exponentials … Fourier Integral Fourier Transform & Properties … Modulation (An application example) Discrete-Time Frequency Domain Methods DT Fourier Series DT Fourier Transform Sampling Theorem Laplace Transform Z Transform

Chapter III Chapter III Fourier Series

Fourier Series Representation of Periodic Signals The objective is to represent complicated signals as linear combination of basic functions, i.e., so that if the response of the LTI system to φ k (t) is known, then the response to x(t) can be expressed as the weighted sum of these responses. Try to select basic functions φ k (t) such that the response to φ k (t) is k φ k (t) φ k (t) are the eigenfunctions of LTI systems k are the eigenvalues of LTI systems Hence, the output is:

Response of LTI Systems to Complex Exponentials Why complex exponential? The most basic periodic signal with a well-defined frequency Complex exponentials are eigenfunctions of LTI systems Response of an LTI system to a complex exponential is the same complex exponential with only change in amplitude CT: e st  H(s) e st DT: z n  H(z) z n Response is scaled version of the input with H(s) or H(z) H(s) and H(z) are complex amplitude factors as functions of complex variables s and z

Response of LTI Systems to Complex Exponentials To find H(s) consider a CT LTI system with h(t). For x(t)= e st For a DT LTI system with h[n], for x[n]= z n  (show as an exercise!) y(t) = H(s) e st, where

Response of LTI Systems to Complex Exponentials Hence, Bottomline: H is known (depends on impulse response) So, if we can express x as a linear combination of complex exponentials (i.e., find a k ), we can write the output in terms of a and H! (No convolution) We mostly use s=jω and z = e jω

Fourier Series of CT Periodic Signals Consider a periodic signal, Then, the fundamental frequency: the fundamental period: minimum positive nonzero value of T Two basic periodic signals: sinusoid : periodic complex exponential : Both periodic with and fundamental frequency of ω 0

Fourier Series of CT Periodic Signals The set of harmonically related complex exponentials: Each of φ k is periodic with fundamental frequency that is an integer multiple of ω 0. All have a common period T 0. Now, take s k =kjω 0 Periodic with T 0 k=0 DC or constant term is the Fourier series representation of x(t). 1 st harmonic (fundamental component) periodic with T 0 2 nd harmonic with fundamental period Need to find a k !!

Fourier Series of CT Periodic Signals Example Consider a periodic signal

Fourier Series of CT Periodic Signals If x(t) is real x*(t)=x(t) and for real periodic signals

Alternative form of Fourier Series Another form:

Determination of Fourier Series of CT Periodic Signals Assume that a given periodic signal x(t) has a FS representation Find FS coefficients, a k of: Multiply both sides with and integrate over one period (from 0 to T=2  /ω 0 )

Determination of Fourier Series of CT Periodic Signals we can take integral over any interval of length T synthesis equationanalysis equation a k : FS coefficients, spectral coefficients a 0 : DC component, average of signal

Determination of Fourier Series of CT Periodic Signals Example Consider the periodic square wave with fundamental period T (ω 0 =2  T) Find FS coefficients! x(t) -T -T/2-T 1 t T1T1 T/2T (average, DC value!)

Determination of Fourier Series of CT Periodic Signals Example

Convergence of Fourier Series FS expansion is possible if x(t) is a periodic function satisfying Dirichlet conditions: C2) x(t) has a finite number of maxima and minima within any period. C3) x(t) has a finite number of discontinuities within a period. C1) Over any period x(t) is absolutely integrable, i.e.,

Convergence of Fourier Series Examples x(t) t Not absolutely integrable over period! Violates C1!

Convergence of Fourier Series Example - meets C1 ! - violates C2! ) violates C3! x(t)

Convergence of Fourier Series Complex exponentials are continuous for. Hence is continuous as well. What if x(t) has discontinuity at t=t 0 ? If Dirichlet conditions are satisfied x(t) is equal to at every continuity point of x(t). At discontinuities, equals to the average of the values on either side of the discontinuity, i.e.,, if x(t) is discontinuous at t=t 0, Remarks:

Properties of Fourier Series 1) Even and Odd Functions: if a function x(t) is even, x(t)=x(-t) a k =a -k Hence, if x(t) is odd a k =-a -k, a 0 =0

Properties of Fourier Series 2) Time Shifting: if x(t) has a k as FS coefficients, i.e.,

Properties of Fourier Series 3. Time Reversal: (prove it as exercise !) 4. Differentiation: … (prove it as exercise !)

Consider the cascade (series) connection of two LTI systems, whose impulse responses are and Evaluate the output signal y[n] corresponding to the input signal