In this lecture you will learn: System impulse response, linear constant-coefficient difference equations, fourier transforms and frequency response. Inviting you refer.
Digital Signal Processing, Fall 2006 - E-Study - Lecture 2: Fourier transforms and frequency response Zheng-Hua Tan Department of Electronic Systems Aalborg University, Denmark zt@kom.aau.dk Digital Signal Processing, II, Zheng-Hua Tan, 2006 Course at a glance MM1 Discrete-time signals and systems MM2 Fourier-domain representation Sampling and reconstruction System System structures System analysis MM6 MM5 Filter MM4 z-transform MM3 DFT/FFT Filter structures MM9,MM10 MM7 Filter design MM8 Digital Signal Processing, II, Zheng-Hua Tan, 2006 Part I: System impulse response System impulse response Linear constant-coefficient difference equations Fourier transforms and frequency response Digital Signal Processing, II, Zheng-Hua Tan, 2006 FIR systems – reflected in the h[n] Ideal delay y[n] = x[n − nd ], − ∞ < n < ∞ h[n] = δ [n − nd ], nd a positive integer Forward difference y[n] = x[n + 1] − x[n] h[n] = δ [n + 1] − δ [n] nd -1 Backward difference y[n] = x[n] − x[n − 1] h[n] = δ [n] − δ [n − 1] Finite-duration impulse response (FIR) system The impulse response has only a finite number of nonzero samples Digital Signal Processing, II, Zheng-Hua Tan, 2006 IIR systems – reflected in the h[n] Accumulator y[n] = n ∑ x[k ] … k = −∞ h[n] = n ∑ δ [k ] = u[n] k = −∞ Infinite-duration impulse response (IIR) system The impulse response is infinitive in duration Stability ? S = ∑ n = −∞ | h[n] | < ∞ ∞ FIR systems always are stable, if each of h[n] values is finite in magnitude IIR systems can be stable, e.g h[n] = a n u[n] with | a |< S = ∑0 | a |n = (1− | a |) < ∞ ∞ Digital Signal Processing, II, Zheng-Hua Tan, 2006 Cascading systems Causality x[n] x[n] Ideal delay ? h[n] = 0, n < h[n] = δ [n − nd ] Forward difference h[n] = δ [ n + 1] − δ [ n] One - sameple delay h[n] = δ [ n − 1] Backward difference h[n] = δ [n] − δ [n − 1] y[n] y[n] Any noncausal FIR system can be made cause by cascading it with a sufficiently long delay! Digital Signal Processing, II, Zheng-Hua Tan, 2006 Cascading systems Accumulator + Backward difference system x[n] Accumulator system h[n] = u[n] x[n] Backward difference h[n] = δ [n] − δ [n − 1] h[n] = δ [n] x[n] x[n] Inverse system: h[n] * hi [n] = hi [n] * h[n] = δ [n] Digital Signal Processing, II, Zheng-Hua Tan, 2006 Part II: LCCD equations System impulse response Linear constant-coefficient difference equations Fourier transforms and frequency response Digital Signal Processing, II, Zheng-Hua Tan, 2006 LCCD equations An important class of LTI systems: input and output satisfy an Nth-order LCCD equations N M k =0 m=0 ∑ ak y[n − k ] = ∑ bm x[n − m] Difference equation representation of the accumulator y[n] = n ∑ x[k ] k = −∞ y[n − 1] = x[n] y[n] n −1 ∑ x[k ] k = −∞ y[n] = x[n] + One-sample delay n −1 ∑ x[k ] = x[n] + y[n − 1] k = −∞ y[n] − y[n − 1] = x[n] y[n-1] Recursive representation Digital Signal Processing, II, Zheng-Hua Tan, 2006 Part III: Fourier transforms System impulse response Linear constant-coefficient difference equations Fourier transforms and frequency response 10 Frequency-domain representation of discrete-time signals and systems Symmetry properties of the Fourier transform Fourier transform theorems Digital Signal Processing, II, Zheng-Hua Tan, 2006 Signal representations A sum of scaled, delayed impulse x[n] = ∞ ∑ x[k ]δ [n − k ] y[n] = k = −∞ k = −∞ ∞ ∑ x[k ]h[n − k ] Sinusoidal and complex exponential sequences Sinusoidal input Ỉ sinusoidal response with the same frequency and with amplitude and phase determined by x[n] = A cos(ω n + φ ) the system Complex exponential sequences are eigenfunctions of LTI systems x[n] = e jωn Ỉ signal representation based on sinusoids or complex exponentials 11 Digital Signal Processing, II, Zheng-Hua Tan, 2006 Eigenfunctions Complex exponentials as input to system h[n] x[n] = e jωn , − ∞ < n < ∞ y[n] = T {e jωn } = ∞ ∑ h[k ]e jω ( n − k ) k = −∞ ∞ = ( ∑ h[k ]e − jωk ) e jωn A – linear operator k = −∞ Define Then H ( e jω ) = ∑ h[k ]e − j ωk k = −∞ y[n] = H (e jω )e jωn Eigenvalue 12 ∞ Eigenfunction Digital Signal Processing, II, Zheng-Hua Tan, 2006 Eigenvalue – called frequency response Frequency response is generally complex H (e jω ) = H R (e jω ) + jH I (e jω ) =| H (e jω ) | e j∠H ( e jω ) describes changes in magnitude and phase Frequency response of the ideal delay system y[n] = x[n − nd ] → h[ n] = δ [n − nd ] Method 1: using the eigenfunction y[n] = T {e jωn } = e jω ( n − nd ) = e − jωnd e jωn H ( e jω ) = e − jω n d ∞ ∑ δ [n − n n = −∞ 13 H I (e jω ) = − sin(ωnd ) Method 2: using the impulse response H ( e jω ) = H R (e jω ) = cos(ωnd ), d | H (e jω ) |= 1, ∠H (e jω ) = −ωnd ]e − jωn =e − jωnd Digital Signal Processing, II, Zheng-Hua Tan, 2006 Frequency response The frequency response of discrete-time LTI systems is always a periodic function of the frequency variable w with period 2π H ( e j ( ω + 2π ) ) = 14 ∞ ∞ n = −∞ n = −∞ ∑ h[n]e − j (ω + 2π ) n = ∑ h[n]e − jωn e − j 2πn = H (e jω ) Only specify over the interval −π < ω ≤ π The ‘low frequencies’ are close to The ‘high frequencies’ are close to ±π Digital Signal Processing, II, Zheng-Hua Tan, 2006 Ideal frequency-selective filters For which the frequency response is unity over a certain range of frequencies, and is zero at the remaining frequencies 15 Ideal low-pass filter: passes only low and rejects high Digital Signal Processing, II, Zheng-Hua Tan, 2006 Ideal frequency-selective filters 16 Digital Signal Processing, II, Zheng-Hua Tan, 2006 Sinusoidal response of LTI systems Sinusoidal input Ỉ sinusoidal response with the same frequency and with amplitude and phase determined by the system x[n] = A cos(ω0 n + φ ) A jφ jω0 n A − jφ − jω0 n e e + e e 2 A jφ A y[ n] = e H (e jω0 )e jω0 n + e − jφ H (e − jω0 )e − jω0 n 2 = if h[n] is real, then H (e − jω0 ) = H * (e jω0 ) =| H (e jω0 ) | e −∠H ( e jω0 ) =| H (e jω0 ) | e −θ then y[ n] = A | H (e jω0 ) | cos(ω0 n + φ + θ ) 17 Digital Signal Processing, II, Zheng-Hua Tan, 2006 Signal representation More than sinusoids, a broad class of signals can be represented as a linear combination of complex exponentials: x[ n] = ∑ α k e jω k n k ∴ y[ n] = ∑ α k H (e jω k )e jω k n k If x[n] can be represented as a superposition of complex exponentials, output y[n] can be computed by using the frequency response, which is similar to the function of impulse response 18 Digital Signal Processing, II, Zheng-Hua Tan, 2006 Frequency-domain representation of x[n] By Fourier transforms Ỉ Fourier representation: x[n] = 2π π ∫−π X (e jω )e jωn dω Inverse Fourier transform w is continuous-time variable ∞ where X (e jω ) = ∑ x[n]e − jωn Fourier transform n is discrete-time variable −∞ represent x[n] as a superposition of infinitesimally small complex sinusoids X (e jω )e jωn dω 2π Fourier spectrum In general, Fourier transform is complex X (e jω ) = X R (e jω ) + jX I (e jω ) =| X (e jω ) | e j∠X ( e 19 jω ) Spectrum Magnitude spectrum Amplitude spectrum Phase spectrum Digital Signal Processing, II, Zheng-Hua Tan, 2006 Frequency and impulse responses Are a Fourier transform pair Recall H ( e jω ) = ∴ h[n] = 2π ∞ ∑ h[n]e − jωn n = −∞ π ∫−π H (e jω )e jωn dω Fourier transform is periodic with period 2π ∞ X (e jω ) = ∑ x[n]e − jωn −∞ 20 Digital Signal Processing, II, Zheng-Hua Tan, 2006 10 Sufficient condition for Fourier transform Condition for the convergence of the infinite sum ∞ | X (e jω ) | = | ∑ x[n]e − jωn | −∞ ∞ ≤ ∑ | x[n] ||e − jωn | −∞ ∞ ≤ ∑ | x[n] | < ∞ −∞ X[n] is absolutely summable, then its Fourier transform exists (sufficient condition) 21 Digital Signal Processing, II, Zheng-Hua Tan, 2006 Example: ideal lowpass filter Frequency response hlp [n] = 2π ωc ∫−ω e j ωn dω = c ⎧ 1, | ω |< ω c H lp (e jω ) = ⎨ ⎩0, ω c