1. Trang chủ
  2. » Kỹ Thuật - Công Nghệ

Lecture Digital signal processing: Lecture 8 - Zheng-Hua Tan

16 50 0

Đang tải... (xem toàn văn)

Tài liệu hạn chế xem trước, để xem đầy đủ mời bạn chọn Tải xuống

THÔNG TIN TÀI LIỆU

Thông tin cơ bản

Định dạng
Số trang 16
Dung lượng 1,35 MB

Nội dung

Lecture 8 - FIR Filter Design include all of the following content: FIR filter design, commonly used windows, generalized linear-phase FIR filter, the Kaiser window filter design method.

Digital Signal Processing, Fall 2006 Lecture 8: FIR Filter Design Zheng-Hua Tan Department of Electronic Systems Aalborg University, Denmark zt@kom.aau.dk Digital Signal Processing, VIII, Zheng-Hua Tan, 2006 Course at a glance MM1 Discrete-time signals and systems MM2 Fourier-domain representation Sampling and reconstruction MM4 z-transform MM3 DFT/FFT System System analysis System structure MM6 MM5 Filter design MM7, MM8 MM9, MM10 Digital Signal Processing, VIII, Zheng-Hua Tan, 2006 FIR filter design Design problem: the FIR system function „ M H ( z ) = ∑ bk z − k k =0 0≤n≤M ⎧b , h[n] = ⎨ n otherwise ⎩0, Start from impulse response directly „ H ( z ) = h[0] + h[1]z −1 + + h[ M ]z − M Find ‰ the degree M and ‰ the filter coefficients h[k] to approximate a desired frequency response Digital Signal Processing, VIII, Zheng-Hua Tan, 2006 Part I: FIR filter design FIR filter design Commonly used windows Generalized linear-phase FIR filter The Kaiser window filter design method „ „ „ „ Digital Signal Processing, VIII, Zheng-Hua Tan, 2006 Lowpass filter – as an example ⎧1, | ω |< ωc H lp (e jω ) = ⎨ ⎩0, ωc < ω ≤ π sin ωc n h[n] = , −∞ < n < ∞ πn „ Ideal lowpass filter „ IIR filter: based on transformations of continuoustime IIR system into discrete-time ones | H c ( jΩ ) | = 1 + (Ω / Ω c ) N poles FIR filter: how? h[n] is non-causal, infinite! „ Digital Signal Processing, VIII, Zheng-Hua Tan, 2006 Design by windowing Desired frequency responses are often piecewiseconstant with discontinuities at the boundaries between bands, resulting in non-causal and infinte impulse response extending from − ∞ to ∞, but „ n → ±∞, hd [n] → So, the most straightforward method is to truncate the ideal response by windowing and timeshifting: ⎧h [n], | n |≤ M g[ n ] = ⎨ d ⎩0, otherwise h[n] = g[n − M ] „ Digital Signal Processing, VIII, Zheng-Hua Tan, 2006 Design by windowing After Champagne & Labeau, DSP Class Notes 2002 Digital Signal Processing, VIII, Zheng-Hua Tan, 2006 Design by rectangular window In general, „ h[n] = hd [n]w[n] For simple truncation, the window is the rectangular window ⎧1, ≤ n ≤ M w[n] = ⎨ ⎩0, otherwise H ( e jω ) = 2π π ∫πH − d (e jθ ) H (e j (ω −θ ) )dθ Digital Signal Processing, VIII, Zheng-Hua Tan, 2006 Convolution process by truncation Fig 7.19 „ w[n] = 1, − ∞ < n < ∞, then what ? W ( e jω ) = ∞ ∑ 2πδ (ω + 2πr ) r = −∞ jω Digital Signal Processing, VIII, Zheng-Hua Tan, 2006 W (e ) should be narrow band Requirements on the window w[n] = 1, − ∞ < n < ∞, W ( e jω ) = ∞ ∑ 2πδ (ω + 2πr ) r = −∞ W (e jω ) should be narrow band „ Requirements W (e jω ) approximates an impulse to faithfully reproduce the desired frequency response ‰ w[n] as short as possible in duration (the order of the filter) to minimize computation in the implementation of the filter Ỉ Conflicting Take the rectangular window as an example ‰ 10 Digital Signal Processing, VIII, Zheng-Hua Tan, 2006 Rectangular window M+1 M=7 constant 4π 11 Digital Signal Processing, VIII, Zheng-Hua Tan, 2006 Part II: Commonly used windows „ „ „ „ 12 FIR filter design Commonly used windows Generalized linear-phase FIR filter The Kaiser window filter design method Digital Signal Processing, VIII, Zheng-Hua Tan, 2006 Standard windows – time domain Rectangular w[n] = ⎧⎨1, ≤ n ≤ M ⎩0, otherwise ⎧2 n / M , ≤ n ≤ M / „ Triangular ⎪ w[n] = ⎨2 − 2n / M , M / ≤ n ≤ M ⎪0, otherwise ⎩ „ Hanning ⎧0.5 − 0.5 cos(2πn / M ), ≤ n ≤ M w[n] = ⎨ otherwise ⎩0, „ Hamming ⎧0.54 − 0.46 cos(2πn / M ), ≤ n ≤ M w[n] = ⎨ otherwise ⎩0, „ Blackman ⎧0.42 − 0.5 cos(2πn / M ) + 0.08 cos(4πn / M ), ≤ n ≤ M w[n] = ⎨ 13 0, Signal Processing, VIII, Zheng-Hua Tan, 2006 otherwise ⎩Digital „ Standard windows – figure „ Fig 7.21 Plotted for convenience In fact, the window is defined only at integer values of n 14 Digital Signal Processing, VIII, Zheng-Hua Tan, 2006 Standard windows – magnitude 15 Digital Signal Processing, VIII, Zheng-Hua Tan, 2006 Standard windows – comparison Magnitude of side lobes vs width of main lobe Independent of M! 16 Digital Signal Processing, VIII, Zheng-Hua Tan, 2006 Part III: Linear-phase FIR filter FIR filter design Commonly used windows Generalized linear-phase FIR filter The Kaiser window filter design method „ „ „ „ 17 Digital Signal Processing, VIII, Zheng-Hua Tan, 2006 Linear-phase FIR systems Generalized linear-phase system „ H (e jω ) = A(e jω )e − jωα + jβ A(e jω ) is a real function of ω , α and β are real constants Causal FIR systems have generalized linear-phase if h[n] satisfies the symmetry condition „ h[ M − n] = h[n], n = 0,1, , M or h[ M − n] = −h[n], n = 0,1, , M 18 Digital Signal Processing, VIII, Zheng-Hua Tan, 2006 Types of GLP FIR filters h[ M − n] = εh[n], n = 0,1, , M ε = or − 19 M even M odd ε =1 Type I Type II ε = −1 Type III Type IV Digital Signal Processing, VIII, Zheng-Hua Tan, 2006 Generalized linear phase FIR filter „ Often aim at designing causal systems with a generalized linear phase (stability is not a problem) ‰ ‰ If the impulse response of the desired filter is symmetric about M/2, hd [ M − n] = hd [n] Choose windows being symmetric about the point M/2 ⎧w[ M − n], ≤ n ≤ M w[n] = ⎨ ⎩0, otherwise W (e jω ) = We (e jω )e − jωM / We (e jω ) is a real, even function of w Ỉ the resulting frequency response will have a generalized linear phase H (e jω ) = Ae (e jω )e − jωM / 20 Ae (e jω ) is real and even Digital Signal Processing, VIII, Zheng-Hua Tan, 2006 10 Linear-phase lowpass filter – an example Desired frequency response ⎧e − jωM / , | ω |< ωc H lp (e jω ) = ⎨ ωc < ω ≤ π ⎩0, sin[ωc (n − M / 2)] hlp [n] = , −∞ < n < ∞ π (n − M / 2) so hlp [ M − n] = hlp [n] apply a symmetric window Ỉ a linear-phase system h[n] = hlp [n]w[n] = 21 sin[ωc (n − M / 2)] w[n] π (n − M / 2) Digital Signal Processing, VIII, Zheng-Hua Tan, 2006 Window method approximations H d (e jω ) = H e (e jω )e − jωM / H e (e jω ) is real and even W (e jω ) = We (e jω )e − jωM / We (e jω ) is real and even H (e jω ) = Ae (e jω )e − jωM / Ae (e jω ) = 22 2π π ∫−π H e (e jθ )We (e j (ω −θ ) )dθ Digital Signal Processing, VIII, Zheng-Hua Tan, 2006 11 Key parameters „ To meet the requirement of FIR filter, choose ‰ ‰ „ 23 Shape of the window Duration of the window Trail and error is not a satisfactory method to design filters Æ a simple formalization of the window method by Kaiser Digital Signal Processing, VIII, Zheng-Hua Tan, 2006 Part IV: Kaiser window filter design „ „ „ „ 24 FIR filter design Commonly used windows Generalized linear-phase FIR filter The Kaiser window filter design method Digital Signal Processing, VIII, Zheng-Hua Tan, 2006 12 The Kaiser window filter design method „ „ An easy way to find the trade-off between the mainlobe width and side-lobe area The Kaiser window ⎧ I [ β (1 − [(n − α ) / α ]2 )1 / ] , 0≤n≤M ⎪ w[n] = ⎨ I (β ) ⎪0, otherwise ⎩ α = M /2 ‰ ‰ 25 I (⋅) is the zeroth-order modified Bessel function of the first kind β ≥ is an adjustable design parameter The length (M+1) and the shape parameter β can be adjusted to trade side-lobe amplitude for main-lobe width (not possible for preceding windows!) Digital Signal Processing, VIII, Zheng-Hua Tan, 2006 The Kaiser window 26 Digital Signal Processing, VIII, Zheng-Hua Tan, 2006 13 Design FIR filter by the Kaiser window Calculate M and β to meet the filter specification The peak approximation error δ is determined by β (Peak error is fixed for other windows) Define A = −20 log10 δ then ‰ A > 50 ⎧0.1102( A − 8.7), ⎪ 0.4 β = ⎨0.5842(A - 21) + 0.07886( A − 21), 21 ≤ A ≤ 50 ⎪0.0, (Rectangular) A < 21 ⎩ ‰ 27 Passband cutoff frequency ω p is determined by: | H (e jω ) |≥ − δ jω Stopband cutoff frequency ω s by: | H (e ) |≤ δ Transition width Δω = ω s − ω p M must satisfy A−8 M = 2.285Δω Digital Signal Processing, VIII, Zheng-Hua Tan, 2006 A lowpass filter Specifications „ Specifications for a discrete-time lowpass filter − 0.01 ≤| H (e jω ) |≤ + 0.01, ≤ ω ≤ ω p = 0.4π | H (e jω ) |≤ 0.001, ω ≥ ω s = 0.6π δ = 0.01 δ = 0.001 28 Digital Signal Processing, VIII, Zheng-Hua Tan, 2006 14 Design the lowpass filter by Kaiser window „ „ „ „ „ Designing by window method indicating δ = δ , we must set δ = 0.001 Transition width Δω = ω s − ω p = 0.2π A = −20 log10 δ = 60 M = 37 The two parameters: β = 5.653, Cutoff frequency of the ideal lowpass filter ω c = (ω s + ω p ) / = 0.5π „ Impulse response ⎧ sin ωc (n − α ) I [ β (1 − [(n − α ) / α ]2 )1 / ] ⋅ , 0≤n≤M ⎪ h[n] = ⎨ π (n − α ) I (β ) ⎪0, otherwise ⎩ 29 Digital Signal Processing, VIII, Zheng-Hua Tan, 2006 Design the lowpass filter What is the group delay? M/2=18.5 30 Digital Signal Processing, VIII, Zheng-Hua Tan, 2006 15 Summary „ „ „ „ 31 FIR filter design Commonly used windows Generalized linear-phase FIR filter The Kaiser window filter design method Digital Signal Processing, VIII, Zheng-Hua Tan, 2006 Course at a glance MM1 Discrete-time signals and systems MM2 Fourier-domain representation Sampling and reconstruction MM4 z-transform MM3 32 DFT/FFT System System analysis System structure MM6 MM5 Filter design MM7, MM8 MM9, MM10 Digital Signal Processing, VIII, Zheng-Hua Tan, 2006 16 ... Take the rectangular window as an example ‰ 10 Digital Signal Processing, VIII, Zheng-Hua Tan, 2006 Rectangular window M+1 M=7 constant 4π 11 Digital Signal Processing, VIII, Zheng-Hua Tan, 2006... at integer values of n 14 Digital Signal Processing, VIII, Zheng-Hua Tan, 2006 Standard windows – magnitude 15 Digital Signal Processing, VIII, Zheng-Hua Tan, 2006 Standard windows – comparison... „ Digital Signal Processing, VIII, Zheng-Hua Tan, 2006 Design by windowing After Champagne & Labeau, DSP Class Notes 2002 Digital Signal Processing, VIII, Zheng-Hua Tan, 2006 Design by rectangular

Ngày đăng: 11/02/2020, 16:15

TỪ KHÓA LIÊN QUAN