Lecture 9 ELE 301: Signals and Systems Prof. Paul Cuff Princeton University Fall 2011-12 Cuff (Lecture 9) ELE 301: Signals and Systems Fall 2011-12 1 / 16 Discrete-time Fourier Transform Represent a Discrete-time signal using δ functions Properties of the Discrete-time Fourier Transform I Periodicity I Time Scaling Property I Multiplication Property Periodic Discrete Duality DFT Constant-Coefficient Difference Equations Cuff (Lecture 9) ELE 301: Signals and Systems Fall 2011-12 2 / 16 Fourier Transform for Discrete-time Signals Z x[n] = X (f )ei2πfndf ; 1 1 X X (f ) = x[n]e−i2πfn: n=−∞ X (f ) is always periodic with period 1. Cuff (Lecture 9) ELE 301: Signals and Systems Fall 2011-12 3 / 16 Continuous-representation of a discrete-time signal 1 X x(t) , x[k]δ(t − k): k=−∞ Notice that 1 ! Z 1 X F[x(t)] = x[k]δ(t − k) e−i2πft dt −∞ k=−∞ 1 X Z 1 = x[k]δ(t − k)e−i2πft dt k=−∞ −∞ 1 X = x[k]e−i2πfk dt: k=−∞ Cuff (Lecture 9) ELE 301: Signals and Systems Fall 2011-12 4 / 16 Properties of the Discrete-time Fourier Transform Inherits properties from continuous-time. Easy Properties: Linearity Conjugation Convolution = Multiplication in frequency domain Parseval's Theorem (integrate over one period) Time shift Properties that require care: Time-scaling Multiplication (circular convolution in frequency) Cuff (Lecture 9) ELE 301: Signals and Systems Fall 2011-12 5 / 16 Time-scaling In continuous time we can scale by an arbitrary real number. In discrete-time we scale only by integers. For an integer k, define x[n=k] if n is a multiple of k; x [n] = k 0 if n is not a multiple of k: xk [n] , X (kf ): Cuff (Lecture 9) ELE 301: Signals and Systems Fall 2011-12 6 / 16 Compressing in time Compressing in time requires decimation. Cuff (Lecture 9) ELE 301: Signals and Systems Fall 2011-12 7 / 16 Discrete Difference What is the Fourier transform of y[n] = x[n] − x[n − 1]? Cuff (Lecture 9) ELE 301: Signals and Systems Fall 2011-12 8 / 16 Dual Derivative Formula The dual to the continuous-time differentiation formula still holds. i nx[n] , X 0(f ): 2π Cuff (Lecture 9) ELE 301: Signals and Systems Fall 2011-12 9 / 16 Accumulation Pn What is the Fourier transform of y[n] = m=−∞ x[m]? (Hint: Inverse of discrete difference) Cuff (Lecture 9) ELE 301: Signals and Systems Fall 2011-12 10 / 16 Parseval's Theorem Theorem: 1 Z 1=2 X 2 2 Ex = jx[n]j = jX (f )j df : n=−∞ −1=2 Cuff (Lecture 9) ELE 301: Signals and Systems Fall 2011-12 11 / 16 Multiplication Property Multiplication in time equates to circular convolution in frequency. Notice that multiplying δ functions is not well defined. This is dual to what we saw in the Fourier series. Cuff (Lecture 9) ELE 301: Signals and Systems Fall 2011-12 12 / 16 Discrete Periodic Duality periodic signal = discrete transform (though not integer f ) discrete signal = periodic transform Cuff (Lecture 9) ELE 301: Signals and Systems Fall 2011-12 13 / 16 Discrete Fourier Transform Notice that a discrete and periodic signal will have a discrete and periodic transform. This is convenient for numerical computation (computers and digital systems). The DFT is (almost) equivalent to the discrete-time Fourier series of the periodic extension. For period N, let 2 x[0] 3 6 x[1] 7 x = 6 7 6 . 7 4 . 5 x[N − 1] Then 2 3 a0 6 a1 7 DFT [x] = 6 7 6 . 7 4 . 5 aN−1 whereCuffak (Lectureare the 9) Fourier seriesELE coefficients.301: Signals and Systems Fall 2011-12 14 / 16 The DFT Matrix Since the Fourier transform is linear, the DFT can be encompassed in a matrix. DFT [x] = Fx: Matlab uses the fast-Fourier-transform algorithm to compute the DFT (using the fft command). Cuff (Lecture 9) ELE 301: Signals and Systems Fall 2011-12 15 / 16 Constant-Coefficient Difference Equations n M X X ak y[n − k] = bk x[n − k]: k=0 k=0 Find the Fourier Transform of the impulse response (the transfer function of the system, H(f )) in the frequency domain. Cuff (Lecture 9) ELE 301: Signals and Systems Fall 2011-12 16 / 16.