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Chapter Three
Discrete - Time Convolutions
Lecture #7
Rediet Million
AAiT, School Of Electrical and Computer Engineering
r...
Introduction
In this chapter we look at methods on how the discrete-time samples,
once obtained, are processed by a specia...
Introduction
Various equivalent forms of the convolution operation will also be
presented. These equivalent forms of convo...
3.1 Block processing methods
The main feature of block processing methods is that data is
collected and processed in block...
Block processing methods
Direct form convolution
Consider a causal FIR filter order M with impulse response
h = {h0, h1, h2...
Block processing methods
Direct form convolution
and use (1) to expand the limits to 0 ≤ m ≤ n ≤ Lx − 1 + m ≤ Lx − 1 + M
⇒...
Block processing methods
Convolution table
A convenient way of expressing the direct form convolution is as
follows convol...
Block processing methods
Convolution table
Example :
Compute the convolution of h(n) and x(n), where
(Rediet Million) DSP-...
Block processing methods
LTI form convolution
This type of convolution uses linearity and time-invariance properties
of th...
Block processing methods
Matrix form of convolution
The convolution equation may also be written in a matrix form as:
y = ...
Block processing methods
Matrix form of convolution
Example :
Compute y when
We may also rewrite the convolution equation ...
Block processing methods
Flip-and slide form of convolution (Graphical form)
Here h(n) is flipped around or reversed and th...
Block processing methods
Overlap-add block convolution method
In many applications considering the input as a single block...
Block processing methods
Transient and steady-state components
For a length-L input and order-M filter,the output sequence ...
Block processing methods
Transient and steady-state components
(Rediet Million) DSP-Lecture #7 April,2018 15 / 18
Block processing methods
Convolution of infinite sequences
Consider the direct form convolution
y(n) =
min(n,M)
m=max(0,n−L...
Block processing methods
Convolution of infinite sequences
Thus, we have
(Rediet Million) DSP-Lecture #7 April,2018 17 / 18
Block processing methods
(#1 ) Class exercises & Assignment
1) Compute the convolution ,y = h ∗ x, of the filter and input
...
Chapter three reading assignment
Sample Processing Methods
Transposed realization
(Rediet Million) DSP-Lecture #7 April,20...
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Digital Signal Processing[ECEG-3171]-Ch1_L07

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Digital Signal Processing[ECEG-3171]-Ch1_L07

  1. 1. Chapter Three Discrete - Time Convolutions Lecture #7 Rediet Million AAiT, School Of Electrical and Computer Engineering rediet.million@aait.edu.et April,2018 (Rediet Million) DSP-Lecture #7 April,2018 1 / 18
  2. 2. Introduction In this chapter we look at methods on how the discrete-time samples, once obtained, are processed by a special class of linear time-invariant discrete-time systems to produce an output. Major practical methods considered are block-processing and sample-processing methods. Block processing method deals with finite-duration blocks of data at a time.Typical applications include -FIR filtering of finite-duration signals by convolution -DFT/FFT spectral computations -Speech analysis and synthesis and image processing Sample processing methods are primarily used in real -time applications such as: -Real-time filtering, -Digital audio effects processing, -Digital control systems, -Adaptive signal processing (Rediet Million) DSP-Lecture #7 April,2018 2 / 18
  3. 3. Introduction Various equivalent forms of the convolution operation will also be presented. These equivalent forms of convolution are: Direct form:Leads to block diagram realization and sample processing algorithm. LTI form : incorporates linearity and time-invariance properties. Matrix form: represents compact vectorial representation of the filtering operation. Flip-and-slide form:Shows clearly input-on and input-off transients and steady-state behavior. Overlap-add block convolution: used whenever the input is extremely long or infinite in duration. (Rediet Million) DSP-Lecture #7 April,2018 3 / 18
  4. 4. 3.1 Block processing methods The main feature of block processing methods is that data is collected and processed in blocks at a time. Consider a finite set of samples of length L representing a finite time record of the input signal x(n). For a sampling interval of T the duration TR of the data record, in seconds, will be TR = (L − 1)T TR = L fs or L = TR = fs Thus, the signal block or vector of length-L is : x = [x0, x1, x2, .., xL−1] (Rediet Million) DSP-Lecture #7 April,2018 4 / 18
  5. 5. Block processing methods Direct form convolution Consider a causal FIR filter order M with impulse response h = {h0, h1, h2, ..., hM} - The length of the impulse response is Lh = M + 1 The response of the order - M FIR filter to a length-Lx input x(n) i.e x = {x0, x1, x2, ..., xLx −1} is obtained,using the direct form convolution as y(n) = m h(m)x(n − m) -Range of h(m) is 0 ≤ m ≤ M.....................(1) -Range of x(n − m) is 0 ≤ n − m ≤ Lx − 1........(2) To determine the range of values of the output index ’n’ we write (2) in the form of m ≤ n ≤ Lx − 1 + m (Rediet Million) DSP-Lecture #7 April,2018 5 / 18
  6. 6. Block processing methods Direct form convolution and use (1) to expand the limits to 0 ≤ m ≤ n ≤ Lx − 1 + m ≤ Lx − 1 + M ⇒The limits for the output index is 0 ≤ n ≤ Lx − 1 + M..........(3) - Thus, Ly = Lx + M and y = {y0, y1, y2, ..., yLx −1+M} For any values of the output index ’n’ in the range (3),we must determine the summation range over ’m’ in the convolution equation - Changing the sign of (2) we obtain −(Lx − 1) ≤ m − n ≤ 0 and adding ’n’ to all sides n − Lx + 1 ≤ m ≤ n..........(4) - ’m’ must satisfy equation (1) and (4) i.e max(0, n − Lx + 1) ≤ m ≤ min(n, M) In the case of an order M FIR filter and a length Lx input ,the direct form of convolution is given as : y(n) = min(n,M) m=max(0,n−Lx +1) h(m)x(n − m) (Rediet Million) DSP-Lecture #7 April,2018 6 / 18
  7. 7. Block processing methods Convolution table A convenient way of expressing the direct form convolution is as follows convolution table form : y(n) = n=i+j h(i)x(j) Each output yn is the sum of all possible products hi xj for which i + j = m.This leads directly to the convolution table. - In the ij-plane, the condition i + j = n represents the nth anti-diagonal.These anti-diagonal entries are summed to form yn. (Rediet Million) DSP-Lecture #7 April,2018 7 / 18
  8. 8. Block processing methods Convolution table Example : Compute the convolution of h(n) and x(n), where (Rediet Million) DSP-Lecture #7 April,2018 8 / 18
  9. 9. Block processing methods LTI form convolution This type of convolution uses linearity and time-invariance properties of the filter to determine the output sequence. For an input sequence x = [x0, x1, x2, x3, x4] and impulse response h(n), we may rewrite the LTI form of convolution as: y(n) = min(n,Lx −1) m=max(0,n−M) x(m)h(n − m) A convolution table type of operation can be formed for the LTI form of convolution where the output values, yn, may be obtained by summing entries column-wise, with h arranged along the row and x arranged along the column. (Rediet Million) DSP-Lecture #7 April,2018 9 / 18
  10. 10. Block processing methods Matrix form of convolution The convolution equation may also be written in a matrix form as: y = Hx where H is an (L + M)xL matrix built out of the filter’s impulse response,h(n). The columns of H are the successively delayed replicas of the impulse response vector h(n). (Rediet Million) DSP-Lecture #7 April,2018 10 / 18
  11. 11. Block processing methods Matrix form of convolution Example : Compute y when We may also rewrite the convolution equation as : y = Xh where X is an (L + M)x(M + 1) matrix (Rediet Million) DSP-Lecture #7 April,2018 11 / 18
  12. 12. Block processing methods Flip-and slide form of convolution (Graphical form) Here h(n) is flipped around or reversed and then slid over the input data sequence. At each time instant, the output sample is obtained by computing the dot product of the filter vector h with M + 1 input samples aligned below it, as shown below: (Rediet Million) DSP-Lecture #7 April,2018 12 / 18
  13. 13. Block processing methods Overlap-add block convolution method In many applications considering the input as a single block may not be practical.A practical approach is to divide the long input into non-overlapping contiguous blocks of practical length, say L samples. Each block is convolved with the order-M filter h(n) producing the output blocks appropriately combined together to get the overall output. y0 = h ∗ x0 y1 = h ∗ x1 y2 = h ∗ x2 y3 = h ∗ x3 Transient and steady-state components (Rediet Million) DSP-Lecture #7 April,2018 13 / 18
  14. 14. Block processing methods Transient and steady-state components For a length-L input and order-M filter,the output sequence can be divided into three sub-ranges for the range of the output time index n 0 ≤ n ≤ L − 1 + M Input-on transient: It takes the filter M time units before it is completely over the nonzero portion of the input sequence. Steady-state: the filter remains completely over the nonzero portion of the input data. Input-off transient: It represent the output after the input is turned off. (Rediet Million) DSP-Lecture #7 April,2018 14 / 18
  15. 15. Block processing methods Transient and steady-state components (Rediet Million) DSP-Lecture #7 April,2018 15 / 18
  16. 16. Block processing methods Convolution of infinite sequences Consider the direct form convolution y(n) = min(n,M) m=max(0,n−L+1) hmxn−m We can obtain the correct summation limits for the following three cases: 1. M → ∞, L < ∞, i.e infinite filter length and finite input length. 2. M < ∞, L → ∞, i.e finite filter length and infinite input length. 3. M → ∞, L → ∞, i.e infinite filter length and infinite input length. In all the above cases, the output index is 0 ≤ n ≤ ∞ and - When M → ∞ ,the upper limit in the sum becomes min(n, M) = n - When L → ∞ ,the lower limit in the sum becomes max(0, n − L + 1) = 0 (Rediet Million) DSP-Lecture #7 April,2018 16 / 18
  17. 17. Block processing methods Convolution of infinite sequences Thus, we have (Rediet Million) DSP-Lecture #7 April,2018 17 / 18
  18. 18. Block processing methods (#1 ) Class exercises & Assignment 1) Compute the convolution ,y = h ∗ x, of the filter and input a. h = [1, 1, 2, 1] x = [1, 2, 1, 1, 2, 1, 1, 1] b. h = [1, 0, 0, 1] x = [1, 1, 2, 2, 2, 2, 1, 1] using the following five methods i. Direct form ii. Convolution table iii.LTI-form iv.Matrix form v. The overlap-add methods of block convolution. 2) An IIR filter has h(n) = (0.25)nu(n).Derive a closed form expressions for the output y(n) when the input is a. a unit step,x(n) = u(n) b. a pulse of finite duration samples,x(n) = u(n) − u(n − 8) c. an alternating step,x(n) = (−2)nu(n) (Rediet Million) DSP-Lecture #7 April,2018 18 / 18
  19. 19. Chapter three reading assignment Sample Processing Methods Transposed realization (Rediet Million) DSP-Lecture #7 April,2018 19 / 18

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