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Bài giảng Xử lý tín hiệu số - Chapter 4: FIR filtering and convolution has contents: Block processing methods, overlap-add block convolution method, sample processing methods, FIR filtering in direct form.
Chapter p FIR filtering and Convolution Ha Hoang Kha, Ph.D.Click to edit Master subtitle style Ho Chi Minh City University of Technology @ Email: hhkha@hcmut.edu.vn CuuDuongThanCong.com https://fb.com/tailieudientucntt Content Block p processingg methods Convolution: direct form, convolution table Convolution: LTI form,, LTI table Matrix form Flip-and-slide form Overlap-add block convolution method Sample processing methods FIR filtering in direct form Ha H Kha CuuDuongThanCong.com FIR Filtering and Convolution https://fb.com/tailieudientucntt Introduction Block processing methods: data are collected and processed in blocks FIR filtering of finite finite-duration duration signals by convolution Fast convolution of long signals which are broken up in short segments DFT/FFT spectrum computations Speech analysis and synthesis Image processing Sample processing methods: the data are processed one at a timewith each input sample being subject to a DSP algorithm which transforms it into an output sample Real-time applications Digitall audio d effects ff processing Digital control systems Adaptive signal processing Ha H Kha CuuDuongThanCong.com FIR Filtering and Convolution https://fb.com/tailieudientucntt Block Processing method The collected signal samples x(n), n=0, 1,…, L-1, can be thought as a block: x=[x [ 0, x1, …, xL-1] The duration of the data record in second: TL=LT Consider a casual FIR filter of order M with impulse response: h=[h0, h1, …, hM] The length (the number of filter coefficients): Lh=M+1 Ha H Kha CuuDuongThanCong.com FIR Filtering and Convolution https://fb.com/tailieudientucntt 11.1 Direct form The convolution in the direct form: y ( n) = ∑ h( m) x ( n − m) m For DSP implementation, p , we must determine The range of values of the output index n The precise range of summation in m Find index n: index of h(m) Æ 0≤m≤M index of x(n-m) x(n m) Æ 0≤n-m≤L-1 0≤n m≤L Ỉ ≤ m ≤ n ≤m+L-1 ≤ M+L-1 ≤ n ≤ M + L −1 Lx=L input samples which is processed by the filter with order M M=L Lx + M yield the output signal y(n) of length L y = L + M Ha H Kha CuuDuongThanCong.com FIR Filtering and Convolution https://fb.com/tailieudientucntt 1Direct form Find index m: index of x(n-m) ( ) index of h(m) Ỉ 0≤m≤M Ỉ 0≤n-m≤L-1 Ỉ n+L-1≤ m ≤ n max ( 0, n − L + 1) ≤ m ≤ ( M, n ) The direct form of convolution is given as follows: y ( n) = min( M , n ) ∑ h( m) x ( n − m) = h ∗ x m = max(0, (0 n − L +1) with ≤ n ≤ M + L − Thus,, y is longer g than the input p x byy M samples p This property p p y follows from the fact that a filter of order M has memory M and keeps each input sample inside it for M time units Ha H Kha CuuDuongThanCong.com FIR Filtering and Convolution https://fb.com/tailieudientucntt Example Consider the case of an order-3 filter and a length of 5-input signal Find the output o t ? h=[h0, h1, h2, h3] x=[x0, x1, x2, x3, x4 ] yy=h*x=[y h x [y0, y1, y2, y3, y4 , y5, y6, y7 ] Ha H Kha CuuDuongThanCong.com FIR Filtering and Convolution https://fb.com/tailieudientucntt 1.2 Convolution table It can be observed that y ( n) = ∑ h(i) x( j ) i, j i+ j =n Convolution table The convolution table is convenient for quick calculation b h by hand d because b iit displays all required operations compactly Ha H Kha CuuDuongThanCong.com FIR Filtering and Convolution https://fb.com/tailieudientucntt Example Calculate the convolution of the following filter and input signals? h=[1, 2, -1, 1], x=[1, 1, 2, 1, 2, 2, 1, 1] Solution: S l i sum of the values along anti-diagonal line yields the output y: y=[1, =[1 3, 3, 5, 3, 7, 4, 3, 3, 0, 1] Note that there are Ly=L+M=8+3=11 output samples Ha H Kha CuuDuongThanCong.com FIR Filtering and Convolution https://fb.com/tailieudientucntt 1.3 LTI Form LTI form of convolution: y ( n ) = ∑ x ( m) h ( n − m) m Consider the filter h=[h0, h1, h2, h3] and the input signal x=[x0, x1, x2, x3, x4 ] Then, the output is given by y (n) = x0 h(n) + x1h(n − 1) + x2 h(n − 2) + x3 h(n − 3) + x4 h(n − 4) We can represent the input and output signals as blocks: Ha H Kha 10 CuuDuongThanCong.com FIR Filtering and Convolution https://fb.com/tailieudientucntt 1.3 Matrix Form It can be b observed b d th thatt H has h th the same entry t along l each h diagonal di l Such a matrix is known as Toeplitz matrix Matrix representations of convolution are very useful in some applications: Image processing Advanced DSP methods such as parametric spectrum estimation and adaptive filtering Ha H Kha 14 CuuDuongThanCong.com FIR Filtering and Convolution https://fb.com/tailieudientucntt Example Using the matrix form to calculate the convolution of the following filter and input signals? h=[1, 2, -1, 1], x=[1, 1, 2, 1, 2, 2, 1, 1] Solution: since Lx=8, M=3 Æ Ly=Lx+M=11, the filter matrix is 11x8 dimensional Ha H Kha 15 CuuDuongThanCong.com FIR Filtering and Convolution https://fb.com/tailieudientucntt 1.4 Flip-and-slide form The output at time n is given by yn = h0 xn + h1 xn −1 + + hM xn − M Flip-and-slide li d lid fform off convolution l i The flip-and-slide form shows clearly the input-on and input-off transient and steady-state steady state behavior of a filter Ha H Kha 16 CuuDuongThanCong.com FIR Filtering and Convolution https://fb.com/tailieudientucntt 1.5 Transient and steady-state behavior M From LTI convolution: y(n) = ∑h(m)x(n − m) = h0 xn + h1xn−1 + + hM xn−M m=0 The output is divided into subranges: Transient T i andd steady-state d fil outputs: filter Ha H Kha 17 CuuDuongThanCong.com FIR Filtering and Convolution https://fb.com/tailieudientucntt 1.6 Overlap-add block convolution method As the input signal is infinite or extremely large, a practical approach is to divide the longg input p into contiguous g non-overlapping pp g blocks of manageable length, say L samples Overlap Overlap-add add block convolution method: Ha H Kha 18 CuuDuongThanCong.com FIR Filtering and Convolution https://fb.com/tailieudientucntt Example Using the overlap-add method of block convolution with each bock length L=3, calculate the convolution of the following filter and input signals? h=[1, 2, -1, 1], x=[1, 1, 2, 1, 2, 2, 1, 1] Solution: The input is divided into block of length L L=33 The output of each block is found by the convolution table: Ha H Kha 19 CuuDuongThanCong.com FIR Filtering and Convolution https://fb.com/tailieudientucntt Example The output of each block is given by Following F ll i from f time i invariant, i i aligning li i the h output blocks bl k according di to theirs absolute timings and adding them up gives the final results: Ha H Kha 20 CuuDuongThanCong.com FIR Filtering and Convolution https://fb.com/tailieudientucntt Sample processing methods The direct form convolution for an FIR filter of order M is given by Introduce the internal states Sample p p processingg algorithm g Fig: Direct form realization off Mth M h order d fil filter Ha H Kha Sample processing methods convenient for real-time real time applications 21 CuuDuongThanCong.com are FIR Filtering and Convolution https://fb.com/tailieudientucntt Example Consider the filter and input given by Using the sample processing algorithm to compute the output and show the input-off transients Ha H Kha 22 CuuDuongThanCong.com FIR Filtering and Convolution https://fb.com/tailieudientucntt Example Ha H Kha 23 CuuDuongThanCong.com FIR Filtering and Convolution https://fb.com/tailieudientucntt Example Ha H Kha 24 CuuDuongThanCong.com FIR Filtering and Convolution https://fb.com/tailieudientucntt Hardware realizations The FIR filtering algorithm can be realized in hardware using DSP chips, for example the Texas Instrument TMS320C25 MAC: Multiplier Accumulator Ha H Kha 25 CuuDuongThanCong.com FIR Filtering and Convolution https://fb.com/tailieudientucntt Hardware realizations The signal processing methods can efficiently rewritten as In modern DSP chips, the two operations can carried out with a single instruction The total processing time for each input sample of Mth order filter: where Tinstr is one instruction cycle in about 30-80 nanoseconds For real-time application, it requires that Ha H Kha 26 CuuDuongThanCong.com FIR Filtering and Convolution https://fb.com/tailieudientucntt Example What is the longest FIR filter that can be implemented with a 50 nsec per instruction DSP chip p p for digital g audio applications pp with sampling p g frequency fs=44.1 kHz ? Solution: Ha H Kha 27 CuuDuongThanCong.com FIR Filtering and Convolution https://fb.com/tailieudientucntt Homework Problems 4.1, 4.2, 4.3, 4.5, 4.15, 4.18 Ha H Kha 28 CuuDuongThanCong.com FIR Filtering and Convolution https://fb.com/tailieudientucntt ... frequency fs =44 .1 kHz ? Solution: Ha H Kha 27 CuuDuongThanCong.com FIR Filtering and Convolution https://fb.com/tailieudientucntt Homework Problems 4. 1, 4. 2, 4. 3, 4. 5, 4. 15, 4. 18 Ha H Kha 28 CuuDuongThanCong.com... https://fb.com/tailieudientucntt 1 .4 Flip-and-slide form The output at time n is given by yn = h0 xn + h1 xn −1 + + hM xn − M Flip-and-slide li d lid fform off convolution l i The flip-and-slide form shows... range of summation in m Find index n: index of h(m) Ỉ 0≤m≤M index of x(n-m) x(n m) Ỉ 0≤n-m≤L-1 0≤n m≤L Ỉ ≤ m ≤ n ≤m+L-1 ≤ M+L-1 ≤ n ≤ M + L −1 Lx=L input samples which is processed by the filter