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EELE 4310: Digital Signal Processing (DSP) Chapter # 4 : Frequency Analysis of Signals (Part Two) Spring, 2012/2013 EELE 4310: Digital Signal Processing (DSP) - Ch.4 Dr. Musbah Shaat 1/13 Outline 1 Frequency Analysis of Discrete-Time Signals EELE 4310: Digital Signal Processing (DSP) - Ch.4 Dr. Musbah Shaat 2/13 Frequency Analysis of Discrete-Time Signals The Fourier Series for Discrete-Time Periodic Signals ... 1 The Fourier series representation of continuous time periodic signal: Can consist of an infinite number of frequency components (possible frequency range −∞ < f < ∞). The frequency spacing between two successive harmonically related frequencies is 1 , where T is the fundamental period. Tp p The frequency range of discrete-time signals is unique over the interval (−π,π) or (0,2π). Adiscrete-time signal of fundamental period N can consist of frequency components separated by 2π radians or f = 1 cycles. N N The Fourier series representation of the discrete time periodic signal will contain at most N frequency components. EELE 4310: Digital Signal Processing (DSP) - Ch.4 Dr. Musbah Shaat 3/13 Frequency Analysis of Discrete-Time Signals The Fourier Series for Discrete-Time Periodic Signals ... 2 Discrete-Time Fourier Series (DTFS) P Synthesis equation : x(t) = N−1c ej2πkn/N Pk=0 k Analysis equation : c = 1 N−1x(n)e−j2πkn/N k N n=0 where N is the fundamental period of the signal. The Fourier series coefficient {ck} form a periodic sequence when extended outside of the range k = 0,1,··· ,N − 1, i.e. c =c k+N k P P c = 1 N−1x(n)e−j2π(k+N)n/N = 1 N−1x(n)e−j2πkn/N = c k+N N n=0 N n=0 k The spectrum of a signal x(n), which is periodic of period N, is a periodic sequence with period N. Consequently, any N consecutive samples of the signal or its spectrum provide a complete description of the signal in the time and frequency domains. EELE 4310: Digital Signal Processing (DSP) - Ch.4 Dr. Musbah Shaat 4/13
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