DSP-Lec 07-Frequency Analysis of Signals and Systems

Chapter p Frequency Analysis of Signals and Systems Ha Hoang Kha, Ph.D.Click to edit Master subtitle style Ho Chi Minh City University of Technology @ Email: hhkha@hcmut.edu.vn ™ Frequency analysis of signal involves the resolution of the signal into its freq frequency enc (sinusoidal) (sin soidal) components components The process of obtaining the spectrum of a given signal using the basic mathematical tools is known w as frequency q y or spectral p analysis y ™ The term spectrum is used when referring the frequency content of a signal g ™ 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 Ha H Kha Frequency analysis of signals and systems ™ The frequency analysis of signals and systems have three major uses in DSP 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 g Ha H Kha Frequency analysis of signals and systems Content Discrete time Fourier transform DTFT Discrete Fourier transform DFT Fast Fourier transform FFT Ha H Kha Transfer functions and Digital Filter Realizations Discrete-time Fourier transform (DTFT) ™ The Fourier transform of the finite-energy discrete-time signal x(n) is defined as: ∞ X (ω ) = ∑ x(n)e − jω n n =−∞ where h ω=2πf/fs f/f ™ The spectrum X(w) is in general a complex-valued function of f frequency: X (ω ) =| X (ω ) | e jθ (ω ) where h θ (ω ) = arg(( X (ω )) with i h -π ≤ θ (ω ) ≤ π ™ | X (ω ) | ™ θ (ω ) Ha H Kha : is the magnitude spectrum : is the phase spectrum Frequency analysis of signals and systems ™ Determine and sketch the spectra of the following signal: a)) x(n) = δ (n) b) x(n) = a nu (n) with |a|