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Lecture 5: Block Processing for FIR FiltersPowerPoint Presentation

Lecture 5: Block Processing for FIR Filters

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Lecture 5: Block Processing for FIR Filters

- Block processing methods
- recorded data: x = {x0 x1 x2 … xL-1}
- length: Lx = L

- system impulse response: h = {h0 h1 h2 h3 … hM}
- length: Lh = M+1

- output data: y = {y0 y1 y2 y3 … yLy-1}
- length: Ly = Lx + Lh - 1

- key question: how do we process h and x to compute y?

- recorded data: x = {x0 x1 x2 … xL-1}

x

y

EE421, Lecture 5

Block Processing

- Convolution tabley(0) = h(0)x(0)y(1) = h(0)x(1) + h(1)x(0)y(2) = h(0)x(2) + h(1)x(1) + h(2)x(0)y(3) = h(0)x(3) + h(1)x(2) + h(2)x(1) + h(3)x(0)y(4) = h(0)x(4) + h(1)x(3) + h(2)x(2) + h(3)x(1) + h(4)x(0)

y(0)

y(1)

y(2)

y(3)

EE421, Lecture 5

y(4)

Block Processing

- Convolution table
- example: x = {1 2 1 3}, h = {4 2 1}

n=0

n=0

y(0) = 4

y(1) = 10

y(2) = 9

y(3) = 16

y(4) = 7

y(5) = 3

y = {4 10 9 16 7 3}

n=0

EE421, Lecture 5

Block Processing

- LTI Formy(n) = … + x(0)h(n) + x(1)h(n-1) + x(2)h(n-2) + x(3)h(n-3) + …

y(0)

y(1)

y(2)

y(3)

y(4)

EE421, Lecture 5

Block Processing

- LTI form
- example: x = {1 2 1 3}, h = {4 2 1}

n=0

n=0

4

10

9

16

7

3

0

y = {4 10 9 16 7 3}

n=0

EE421, Lecture 5

Block Processing

- Matrix form
- example: x = {1 2 1 3}, h = {4 2 1}

n=0

n=0

y = {4 10 9 16 7 3}

n=0

EE421, Lecture 5

h(2) h(1) h(0)

h(2) h(1) h(0)

h(2) h(1) h(0)

h(2) h(1) h(0)

h(2) h(1) h(0)

y(0)

y(1)

y(2)

y(3)

y(4)

y(5)

Block Processing- Flip-and-slide form:y(n) = h(0)x(n) + h(1)x(n-1) + h(2)x(n-2) + … + h(L)x(n-L)

EE421, Lecture 5

1 2 4

1 2 4

1 2 4

1 2 4

1 2 4

4

10

9

16

7

3

Block Processing- Flip-and-slide form:
- example: x = {1 2 1 3}, h = {4 2 1}

n=0

n=0

y = {4 10 9 16 7 3}

n=0

EE421, Lecture 5

Block Processing

- Programming considerations
- convolution table formfunction y = conv(h, x)
Lh = length(h);Lx = length(x);Ly = Lh+Lx-1;y = zeros(1,Ly);for n = 0:Ly-1 y(n+1) = 0; for i = 0:Lh-1 for j = 0:Lx-1 if (i+j == n) y(n+1) = y(n+1) + h(i+1)*x(j+1); end; end endend

- convolution table formfunction y = conv(h, x)

All these for loops will

be inefficient in Matlab!

EE421, Lecture 5

Block Processing

- Programming considerations
- LTI formfunction y = conv(h, x)Lh = length(h);Lx = length(x);Ly = Lh+Lx-1; y = zeros(1,Ly); for j = 0:Lx-1 shifth = [zeros(1, j) h zeros(1,Ly-Lh-j)]; y = y + x(j+1)*shifth;end

Only 1 for loop!

EE421, Lecture 5

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