2. بسم الله الرحمن الرحيم Digital Signal Processing Lecture 3 Review of Discerete time Fourier Transform (DTFT) University of Khartoum Department of Electrical and Electronic Engineering Diploma/M. Sc. Program in Telecommunication and Information Systems 2013-2014 Dr. Iman AbuelMaaly

3. 2013 -2014 2 Signals Discrete-time Continuous-time Periodic Aperiodic Periodic Aperiodic Frequency Domain Fourier Transform Fourier Series Fourier Transform Fourier Series

4. Fourier Transform of DT Aperiodic Signal The Fourier transform pair for DT signals:The Fourier transform pair for DT signals: Synthesis equation Inverse Transform Analysis Equation Direct Transform 2013 -2014 3

5. Convergence of the DT Fourier transform (DTFT) Uniform convergence is guaranteed if x(n) is absolutely summable. Some sequences are not absolutely summable but square summable having finite energy, Which is a weaker condition for convergence of DF transform. 2013 -2014 4

6. The Gibbs Phenomenon Direct truncation of impulse response leads to well known Gibbs phenomenon. It manifests itself as a fixed percentage overshoot and ripple before and after discontinuity in the frequency response. 2013 -2014 5

7. 6 Example: Gibbs phenomenon illustration Next figures: Magnitude responses of the N- th order FIR low-pass digital filters with normalized cut off frequency, for N=5, 25, 50, 100. The figures confirm the above given statements concerning the Gibbs phenomenon. The Gibbs Phenomenon 2013 -2014

8. 7 Low-Pass FIR Filter: Rectangular Window Application 2013 -2014

9. Energy Density Spectrum of DT signals The energy of a DT signal is defined as The energy relation between x(n) and X(ω) is given by This is the Parseval’s relation for DT aperiodic signal. 2013 -2014 8

10. Energy Density Spectrum of DT signals The quantity Represents the distribution of frequency and it is called the energy density spectrum of x(n). S xx (ω) contains no phase information. 2013 -2014 9

11. Relationship of Fourier Transform to the Z-transform Definition: The Z – transform of a discrete-time signal x(n) is defined as the power series: 2013 -2014 10

12. Relationship of Fourier Transform to the Z- transform Let us express the complex variable z in polar form as: If |z | =1 then z = e jω This yields Then the Z transform can be viewed as the Fourier transform of the sequence evaluated on the unit circle. If the unit circle is not contained in the ROC of X(z) then the Fourier transform does not exist. 2013 -2014 11

13. The Cepstrum Let us consider a sequence {x(n)} having a Z- transform X(z). Assume {x(n)} is a stable sequence so that X(z) convergence on the unit cirlce. {c x (n)} The complex cepstrum of the sequence {x(n)} is defined as {c x (n)} which is the inverse of {C x (z)} where, 2013 -2014 12

14. The Cepstrum Then, Where, Where C is the closed contour about the origin and lies within the ROC. The FT relation: 2013 -2014 13

15. 14 TerminologyTerminology SpectrumSpectrum – Fourier transform of signal CepstrumCepstrum – Inverse Fourier transform of log spectrum FilteringFiltering – operations on spectrum LifteringLiftering – operations on cepstrum FrequencyFrequency – basic unit of spectrum QuefrencyQuefrency – basic unit of cepstrum AnalysisAnalysis – operation on spectrum AlanysisAlanysis – operation on cepstrum 2013 -2014

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17. Properties of Discrete Time Fourier Transform (DTFT) 1.Symmetry Property 2.Linearity 3.Time-Shifting 4.Time-reversal 5.Convolution Theorem 6.The Correlation Theorem 2013 -2014 16 7. The Wiener- Khinchine 8.Frequency Shifting 9.The Modulation Theorem 10.Parseval’s Theorem 11.Windowing Theorem 12.Differentiation in the Frequency Domain

18. The Convolution Theory If And then 2013 -2014 17

19. The Correlation Theorem 2013 -2014 18 If And then

20. The Wiener- Khinchine The energy spectral density of an energy signal is the Fourier transform of its autocorrelation sequence. 2013 -2014 19

21. Parseval’s Theorem If and Then In a special case 2013 -2014 20

22. The Frequency Response Function The Frequency Response Function The response of a relaxed LTI to an arbitrary input signal x(n) is given by, The system is excited by A= amplitude and ω= arbitrary frequency 2013 -2014 21 Eigen valueEigen function

23. 2013 -2014 22 Even function Odd function The Frequency Response Function

24. So, if we know we also know these functions for 2013 -2014 23 Even function Odd function

25. Text Book: John G. Proakis, and D. Manolakis, Digital Signal Processing: Principles, Algorithms and Applications, This lecture covered the following: 2013 -2014 24