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 Dr. Iman AbuelMaaly

Signals Discrete-time Continuous-time Periodic Aperiodic Periodic Aperiodic Frequency Domain Fourier Transform Fourier Series Fourier Transform Fourier Series

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

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

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

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

7 Low-Pass FIR Filter: Rectangular Window Application

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

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

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:

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

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,

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

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

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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 The Wiener- Khinchine 8.Frequency Shifting 9.The Modulation Theorem 10.Parseval’s Theorem 11.Windowing Theorem 12.Differentiation in the Frequency Domain

The Convolution Theory If And then

The Correlation Theorem If And then

The Wiener- Khinchine The energy spectral density of an energy signal is the Fourier transform of its autocorrelation sequence

Parseval’s Theorem If and Then In a special case

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 Eigen valueEigen function

Even function Odd function The Frequency Response Function

So, if we know we also know these functions for Even function Odd function

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