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Course Outline (Tentative)

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 …

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Course Outline (Tentative)

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  1. 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

  2. Chapter IIIFourier Series

  3. 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 toφk(t) is known, then the response to x(t) can be expressed as the weighted sum of these responses. • Try to select basic functionsφk(t)such that the response toφk(t) is kφk(t) • φk(t) are the eigenfunctions of LTI systems • kare the eigenvalues of LTI systems • Hence, the output is:

  4. 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: est H(s) est • DT: zn  H(z) zn • 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

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

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

  7. Both periodic with and fundamental frequency ofω0 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 :

  8. Fourier Series of CT Periodic Signals The set of harmonically related complex exponentials: Each of φk is periodic with fundamental frequency that is an integer multiple of ω0. All have a common period T0. is the Fourier series representation of x(t). Now, take sk=kjω0 Periodic with T0 Need to find ak !! k=0 DC or constant term 1st harmonic (fundamental component) periodic with T0 2nd harmonic with fundamental period

  9. Fourier Series of CT Periodic SignalsExample Consider a periodic signal

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

  11. Alternative form of Fourier Series Another form:

  12. Multiply both sides with and integrate over one period (from 0 to T=2/ω0) Determination of Fourier Series of CT Periodic Signals Assume that a given periodic signal x(t) has a FS representation Find FS coefficients, ak of:

  13. Determination of Fourier Series of CT Periodic Signals we can take integral over any interval of length T synthesis equation analysis equation ak: FS coefficients, spectral coefficients a0: DC component , average of signal

  14. x(t) - - - - - - t -T/2 -T1 T1 T/2 T -T Determination of Fourier Series of CT Periodic SignalsExample Consider the periodic square wave with fundamental period T (ω0=2T) Find FS coefficients! (average, DC value!)

  15. Determination of Fourier Series of CT Periodic SignalsExample

  16. Convergence of Fourier Series FS expansion is possible if x(t) is a periodic function satisfying Dirichlet conditions: C1) Over any period x(t) is absolutely integrable, i.e., 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.

  17. x(t) - - - - - - t -2 1 -1 2 Convergence of Fourier SeriesExamples Not absolutely integrable over period! Violates C1!

  18. - - - 1 2 x(t) - - - Convergence of Fourier SeriesExample - meets C1 ! - violates C2! 3) violates C3!

  19. Convergence of Fourier Series Remarks: • Complex exponentials are continuous for . Hence is continuous as well. • What if x(t) has discontinuity at t=t0 ? • 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=t0,

  20. Properties of Fourier Series 1) Even and Odd Functions: if a function x(t) is even, x(t)=x(-t) ak=a-k Hence, ak=-a-k , a0=0 if x(t) is odd

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

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

  23. 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

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