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Signals and Systems Fall 2003 Lecture #6 23 September 2003

Signals and Systems Fall 2003 Lecture #6 23 September 2003. 1. CT Fourier series reprise, properties, and examples 2. DT Fourier series 3. DT Fourier series examples anddifferences with CTFS. CT Fourier Series Pairs. Review:. Skip it in future for shorthand.

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Signals and Systems Fall 2003 Lecture #6 23 September 2003

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  1. Signals and Systems Fall 2003 Lecture #6 23 September 2003 1. CT Fourier series reprise, properties, and examples 2. DT Fourier series 3. DT Fourier series examples anddifferences with CTFS

  2. CT Fourier Series Pairs Review: Skip it in future for shorthand

  3. Another (important!) example: Periodic Impulse Train Sampling function important for sampling • ─ All components have: • the same amplitude, • & • the same phase.

  4. (A few of the) Properties of CT Fourier Series • Linearity • Conjugate Symmetry • Time shift Introduces a linear phase shift ∝to

  5. Example: Shift by half period using

  6. • Parseval’s Relation Power in the kth harmonic Average signal poser Energy is the same whether measured in the time-domain or the frequency-domain • Multiplication Property (Both x(t) and y(t) are periodic with the same period T) Proof:

  7. Periodic Convolutionx(t), y(t) periodic with period T Not very meaningful E.g. If both x(t) and y(t) are positive, then

  8. Periodic Convolution (continued) Periodic convolution:Integrate over anyone period (e.g. -T/2 to T/2) where otherwise

  9. Periodic Convolution (continued) Facts 1) z(t) is periodic with period T (why?) 2) Doesn’t matter what period over which we choose to integrate: Periodic 3) Convolution in time Multiplication In frequency!

  10. Fourier Series Representation of DT Periodic Signals • x[n] -periodic with fundamental period N, fundamental frequency • Only e jωnwhich are periodic with period N will appear in the FS • There are only N distinct signals of this form • So we could just use • However, it is often useful to allow the choice of N consecutive values of k to be arbitrary.

  11. DT Fourier Series Representation = Sum over any N consecutive values of k — This is a finite series - Fourier (series) coefficients Questions: 1) What DT periodic signals have such a representation? 2) How do we find ak?

  12. Answer to Question #1: Any DT periodic signal has a Fourier series representation N equations for N unknowns, a0, a1, …, a N-1

  13. A More Direct Way to Solve for ak Finite geometric series otherwise

  14. So, from multiply both sides by and then orthoronality

  15. DT Fourier Series Pair (Synthesis equation) (Analysis equation) Note:It is convenient to think of akas being defined for all integers k. So: 1) ak+N= ak —Special property of DT Fourier Coefficients. 2) We only use Nconsecutive values of ak in the synthesis equation. (Since x[n] is periodic, it is specified by N numbers, either in the time or frequency domain)

  16. Example #1: Sum of a pair of sinusoids periodic with period

  17. Example #2: DT Square Wave multiple of Using

  18. Example #2: DT Square Wave (continued)

  19. Convergence Issues for DT Fourier Series: Not an issue, since all series are finite sums. Properties of DT Fourier Series: Lots, just as with CT Fourier Series Example: Frequency shift

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