Lecture Digital signal processing: Chapter 7 - Nguyen Thanh Tuan

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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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Nội dung Text: Lecture Digital signal processing: Chapter 7 - Nguyen Thanh Tuan

Chapter 7 Frequency 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 FFT Digital 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 spectrum Digital 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 x1 (n)  F  X1 ( ) x2 (n)  F  X 2 ( ) then a1 x1 (n)  a2 x2 (n)  F  a1 X1 ()  a2 X 2 ()  Time-shifting: if x(n)  F  X ( ) then x(n  k )  F  e jk X ( ) Digital Signal Processing 9 Frequency analysis of signals and systems
Properties of DTFT  Time reversal: if x(n)  F  X ( ) then x(n)  F  X ( )  Convolution theory: if x1 (n)  F  X1 ( ) x2 (n)  F  X 2 ( ) then x(n)  x1 (n)  x2 (n)  F  X ()  X1 () X 2 () Example: Using DTFT to calculate the convolution of the sequences x(n)=[1 2 3] and h(n)=[1 0 1]. Digital Signal Processing 10 Frequency analysis of signals and systems
Frequency resolution and windowing  The duration of the data record is:  The rectangular window of length L is defined as:  The windowing processing has two major effects: reduction in the frequency resolution and frequency leakage. Digital Signal Processing 11 Frequency analysis of signals and systems
Rectangular window Digital Signal Processing 12 Frequency analysis of signals and systems
Impact of rectangular window  Consider a single analog complex sinusoid of frequency f1 and its sample version:  With assumption , we have Digital Signal Processing 13 Frequency analysis of signals and systems
Double sinusoids  Frequency resolution: Digital Signal Processing 14 Frequency analysis of signals and systems
Hamming window Digital Signal Processing 15 Frequency analysis of signals and systems
Non-rectangular window  The standard technique for suppressing the sidelobes is to use a non- rectangular window, for example Hamming window.  The main tradeoff for using non-rectangular window is that its mainlobe becomes wider and shorter, thus, reducing the frequency resolution of the windowed spectrum.  The minimum resolvable frequency difference will be where : c=1 for rectangular window and c=2 for Hamming window. Digital Signal Processing 16 Frequency analysis of signals and systems
Example  The following analog signal consisting of three equal-strength sinusoids at frequencies where t (ms), is sampled at a rate of 10 kHz. We consider four data records of L=10, 20, 40, and 100 samples. They corresponding of the time duarations of 1, 2, 4, and 10 msec.  The minimum frequency separation is Applying the formulation , the minimum length L to resolve all three sinusoids show be 20 samples for the rectangular window, and L =40 samples for the Hamming case. Digital Signal Processing 17 Frequency analysis of signals and systems
Example Digital Signal Processing 18 Frequency analysis of signals and systems
Example Digital Signal Processing 19 Frequency analysis of signals and systems
2. Discrete Fourier transform (DFT)  X ( ) is a continuous function of frequency and therefore, it is not a computationally convenient representation of the sequence x(n).  DFT will present x(n) in a frequency-domain by samples of its spectrum X ( ) .  A finite-duration sequence x(n) of length L has a Fourier transform: L 1 X ( )   x(n)e jn 0    2 n 0 Sampling X(ω) at equally spaced frequency k  2 k , k=0, 1,…,N-1 where N ≥ L, we obtain N-point DFT of length N L-signal: 2 k L 1 X (k )  X ( )   x(n)e j 2 kn / N (N-point DFT) N n 0  DFT presents the discrete-frequency samples of spectra of discrete- time signals. Digital Signal Processing 20 Frequency analysis of signals and systems