Lecture Digital signal processing: Chapter 7 - Nguyen Thanh Tuan

After studying this chapter students will be able to understand frequency analysis of signals and systems. This chapter includes content: Discrete time fourier transform DTFT, discrete fourier transform DFT, fast fourier transform FFT.
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Lecture Digital signal processing: Chapter 7 - Nguyen Thanh TuanChapter 7Frequency Analysis of Signals and Systems Nguyen Thanh Tuan, Click M.Eng. to edit Master subtitle style Department of Telecommunications (113B3) Ho Chi Minh City University of Technology Email: nttbk97@yahoo.com  Frequency analysis of signal involves the resolution of the signal into its frequency (sinusoidal) components. The process of obtaining the spectrum of a given signal using the basic mathematical tools is known as frequency or spectral analysis.  The term spectrum is used when referring the frequency content of a signal.  The process of determining the spectrum of a signal in practice base on actual measurements of signal is called spectrum estimation.  The instruments of software programs used to obtain spectral estimate of such signals are kwon as spectrum analyzers.Digital Signal Processing 2 Frequency analysis of signals and systems  The frequency analysis of signals and systems have three major uses in DSP: 1) The numerical computation of frequency spectrum of a signal. 2) The efficient implementation of convolution by the fast Fourier transform (FFT) 3) The coding of waves, such as speech or pictures, for efficient transmission and storage.Digital Signal Processing 3 Frequency analysis of signals and systems Content 1. Discrete time Fourier transform DTFT 2. Discrete Fourier transform DFT 3. Fast Fourier transform FFTDigital Signal Processing 4 Transfer functions and Digital Filter Realizations 1. Discrete-time Fourier transform (DTFT)  The Fourier transform of the finite-energy discrete-time signal x(n) is defined as:  X ( )   x(n)e jn n  where ω=2πf/fs  The spectrum X(w) is in general a complex-valued function of frequency: X ( ) | X () | e j ( ) where  ()  arg( X ()) with -   ()    | X ( ) | : is the magnitude spectrum   ( ) : is the phase spectrumDigital Signal Processing 5 Frequency analysis of signals and systems  Determine and sketch the spectra of the following signal: a) x(n)   (n) b) x(n)  a nu(n) with |a| Inverse discrete-time Fourier transform (IDTFT)  Given the frequency spectrum X ( ) , we can find the x(n) in time- domain as  1 x ( n)  2   X ( )e jn d which is known as inverse-discrete-time Fourier transform (IDTFT) Example: Consider the ideal lowpass filter with cutoff frequency wc. Find the impulse response h(n) of the filter.Digital Signal Processing 7 Frequency analysis of signals and systems Properties of DTFT  Symmetry: if the signal x(n) is real, it easily follows that X  ( )  X ( ) or equivalently, | X () || X () | (even symmetry) arg( X ())   arg( X ()) (odd symmetry) We conclude that the frequency range of real discrete-time signals can be limited further to the range 0 ≤ ω≤π, or 0 ≤ f≤fs/2.  Energy density of spectrum: the energy relation between x(n) and X(ω) is given by Parseval’s relation:   1 E x   | x ( n) |   X ( ) d 2 2 n  2  S xx ( ) | X ( ) |2 is called the energy density spectrum of x(n)Digital Signal Processing 8 Frequency analysis of signals and systems Properties of DTFT  The relationship of DTFT and z-transform: if X(z) converges for  |z|=1, then X ( z ) |z e    x(n)e jn  X ( ) j n   Linearity: if ...