DSP-Chapter4 Student 11012016

8/18/2019 DSP-Chapter4 Student 11012016

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Nguyen Thanh Tuan, M.Eng.

Department of Telecommunications (113B3)

Ho Chi Minh City University of Technology

Email: [email protected]

FIR filtering and Convolution

Chapter 4


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Digital Signal Processing

Content

2 FIR Filtering and Convolution

Block processing 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


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Introduction

3

Block processing methods: data are collected and processed in blocks.

FIR Filtering and Convolution

FIR filtering of finite-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 time- with each input sample being subject to a DSP algorithm whichtransforms it into an output sample.

Real-time applications

Digital audio effects processing

Digital control systems

Adaptive signal processing


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1. Block Processing method

4

The collected signal samples x(n), n=0, 1,…, L-1, can be thought as a

block:

The duration of the data record in second: TL=LT

x=[x0, x1, …, xL-1 ]

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



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11.1. Direct form

5

The convolution in the direct form:

Find index n: index of h(m) 0≤m≤M

( ) ( ) ( )m

y n h m x n m

For DSP implementation, we must determine

The range of values of the output index n

The precise range of summation in m

index of x(n-m) 0≤n-m≤L-1

0 ≤ m ≤ n ≤m+L-1 ≤ M+L-10 n M L 1

Lx=L input samples which is processed by the filter with order Myield the output signal y(n) of length yL L M=L M x



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1Direct form

6

Find index m: index of h(m) 0≤m≤M

index of x(n-m) 0≤n-m≤L-1 n+L-1≤ m ≤ n

max 0, n L 1 m min M, n

min( , )

max(0, 1)

( ) ( ) ( ) M n

m n L

y n h m x n m

h x

The direct form of convolution is given as follows:


0 n M L 1 with

Thus, y is longer than the input x by M samples. This propertyfollows from the fact that a filter of order M has memory M andkeeps each input sample inside it for M time units.


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Example 1

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Consider the case of an order-3 filter and a length of 5-input signal.Find the output ?


h=[h0, h1, h2, h3 ]

x=[x0, x1, x2, x3, x4 ]

y=h*x=[y 0, y 1, y 2, y 3, y 4 , y 5, y 6, y 7 ]


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1.2. Convolution table

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It can be observed that


,

( ) ( ) ( )i j

i j n

y n h i x j

Convolution table

The convolutiontable is convenientfor quick calculationby hand because it

displays all requiredoperationscompactly.


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Example 2

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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:

sum of the values along anti-diagonal line yields the output y:

y=[1, 3, 3, 5, 3, 7, 4, 3, 3, 0, 1]

Note that there are Ly =L+M=8+3=11 output samples.


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1.3. LTI Form

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LTI form of convolution:


( ) ( ) ( )m

y n x m h n m

0 1 2 3 4( ) ( ) ( 1) ( 2) ( 3) ( 4) y n x h n x h n x h n x h n x h n

Consider the filter h=[h0, h1, h2, h3 ] and the input signal x=[x0, x1, x2,x3, x4 ]. Then, the output is given by

We can represent the input and output signals as blocks:


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1.3. LTI Form


LTI form of convolution:

LTI form of convolution provides a more intuitive way to understand the linearity and time-invariance properties of the filter.


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Example 3


Using the LTI form to calculate the convolution of the followingfilter and input signals?

h=[1, 2, -1, 1], x=[1, 1, 2, 1, 2, 2, 1, 1]

Solution:


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1.4. Matrix Form


Based on the convolution equations

y Hx

x is the column vector of the Lx input samples.

y is the column vector of the Ly =Lx+M put samples.

H is a rectangular matrix with dimensions (Lx+M)xLx .

we can write


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1.4. Matrix Form


It can be observed that H has the same entry along each diagonal.Such a matrix is known as Toeplitz matrix.

Matrix representations of convolution are very useful in someapplications:

Image processing

Advanced DSP methods such as parametric spectrum estimation and adaptive

filtering


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Example 4


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 Ly =Lx+M=11, the filter matrix is11x8 dimensional


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1.5. Flip-and-slide form


0 1 1 ...n n n M n M y h x h x h x

The output at time n is given by

Flip-and-slide form of convolution

The flip-and-slide form shows clearly the input-on and input-offtransient and steady-state behavior of a filter.


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1.6. Transient and steady-state behavior


Transient and steady-state filter outputs:

From LTI convolution:0 1 1

0

( ) ( ) ( ) ... M

n n M n M

m

y n h m x n m h x h x h x

The output is divided into 3 subranges:


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1.7. Overlap-add block convolution method


Overlap-add block convolution method:

As the input signal is infinite or extremely large, a practical approach

is to divide the long input into contiguous non-overlapping blocks ofmanageable length, say L samples.


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Example 5


Using the overlap-add method of block convolution with each bock

length L=3, calculate the convolution of the following filter andinput 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=3

The output of each block is found by the convolution table:


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Example 5


The output of each block is given by

Following from time invariant, aligning the output blocks accordingto theirs absolute timings and adding them up gives the final results:


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2. Sample processing methods


The direct form convolution for an FIR filter of order M is given by

Fig: Direct form realizationof Mth order filter

Sample processing algorithm

Introduce the internal states

Sample processing methods areconvenient for real-time applications


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Example 6


Consider the filter and input given by

Using the sample processing algorithm to compute the output andshow the input-off transients.


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Example 6



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Example



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Hardware realizations


The FIR filtering algorithm can be realized in hardware using DSP

chips, for example the Texas Instrument TMS320C25

MAC: Multiplier Accumulator


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Hardware realizations


The signal processing methods can efficiently rewritten as

In modern DSP chips, the twooperations

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


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Example 7


What is the longest FIR filter that can be implemented with a 50 nsec

per instruction DSP chip for digital audio applications with samplingfrequency f s=44.1 kHz ?

Solution:


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Homework 1



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Homework 2



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Homework 4


Compute the output y(n) of the filter h(n) = {1, -1, 1, -1} and input

x(n) = {1, 2, 3, 4, @, -3, 2, -1}


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Homework 5

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Compute the convolution, y = h ∗ x, of the filter and input,

h(n) = {1, -1, -1, 1} , x(n) = {1, 2, 3, 4, @, -3, 2, -1} using thefollowing methods:1. The convolution table.2. The LTI form of convolution, arranging the computations in a

table form.3. The overlap-add method of block convolution with length-3

input blocks.4. The overlap-add method of block convolution with length-4

input blocks.5. The overlap-add method of block convolution with length-5

input blocks.


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