Lecture 5: Block Processing for FIR Filters

1
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 M1
• 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?

x
y
2
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)
y(4)
3
Block Processing

• Convolution table
• example x 1 2 1 3, h 4 2 1

n0
n0
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
n0
4
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)
5
Block Processing

• LTI form
• example x 1 2 1 3, h 4 2 1

n0
n0
4
10
9
16
7
3
0
y 4 10 9 16 7 3
n0
6
Block Processing

• Matrix form

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Block Processing

• Matrix form
• example x 1 2 1 3, h 4 2 1

n0
n0
y 4 10 9 16 7 3
n0
8
Block Processing

• Flip-and-slide formy(n) h(0)x(n)
  h(1)x(n-1) h(2)x(n-2) h(L)x(n-L)

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Block Processing

• Flip-and-slide form
• example x 1 2 1 3, h 4 2 1

n0
n0
y 4 10 9 16 7 3
n0
10
Block Processing

• Programming considerations
• convolution table formfunction y conv(h, x)
• Lh length(h)Lx length(x)Ly LhLx-1y
  zeros(1,Ly)for n 0Ly-1 y(n1) 0 for
  i 0Lh-1 for j 0Lx-1 if (ij n)
  y(n1) y(n1) h(i1)x(j1) end end
  endend

All these for loops will be inefficient in Matlab!
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Block Processing

• Programming considerations
• LTI formfunction y conv(h, x)Lh
  length(h)Lx length(x)Ly LhLx-1 y
  zeros(1,Ly) for j 0Lx-1 shifth zeros(1,
  j) h zeros(1,Ly-Lh-j) y y
  x(j1)shifthend

Only 1 for loop!