Signals and Systems

Lecture 19: FIR Filters

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Today's lecture −System Properties:

Linearity Time-invariance

−How to convolve the signals−LTI Systems characteristics−Cascade LTI Systems

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Time Invariance

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Testing Time-Invariance

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Examples of Time-Invariance−Square Law system

y[n] = {x[n]} 2

−Time Flip systemy[n] = x[- n]

−First Difference system y[n] = x[n] - x[n-1]

−Practice: Prove the system given Exercise 5.9 is not time-invariant

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Linear System

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Testing Linearity

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Practice Problems−y[n] = x[n - 2] – 2 x[n] + x[n + 2]−y[n] = x[n] cos(0.2n)−y[n] = n x[n]

−Are all FIR filters Time-invariant and Linear?

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There are two methods to convolve the signals:

− Graphical Method− Tabular Method

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Convolution− Example

knhkxnyk

5 nununx

41 nununh

0 1 2 3 4123 k

1

knhkx

ny

0 1 2 3 4 5n

6 7

kh kx

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Convolution− Example

knhkxnyk

5 nununx

41 nununh

0 1 2 3 4123 k

1

knhkx

ny

0 1 2 3 4 5n

6 7

1

12

Convolution− Example

knhkxnyk

5 nununx

41 nununh

0 1 2 3 4123 k

1

knhkx

ny

0 1 2 3 4 5n

6 7

12

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Convolution− Example

knhkxnyk

5 nununx

41 nununh

0 1 2 3 4123 k

1

knhkx

ny

0 1 2 3 4 5n

6 7

12

3

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Convolution− Example

knhkxnyk

5 nununx

41 nununh

0 1 2 3 4123 k

1

knhkx

ny

0 1 2 3 4 5n

6 7

12

3 3

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Convolution− Example

knhkxnyk

5 nununx

41 nununh

0 1 2 3 4123 k

1

knhkx

ny

0 1 2 3 4 5n

6 7

12

3 3 3

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Convolution− Example

knhkxnyk

5 nununx

41 nununh

0 1 2 3 4123 k

1

knhkx

ny

0 1 2 3 4 5n

6 7

12

3 3 3

5

2

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Convolution− Example

knhkxnyk

5 nununx

41 nununh

0 1 2 3 4123 k

1

knhkx

ny

0 1 2 3 4 5n

6 7

12

3 3 3

5

2

6

1

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Convolution− Example

knhkxnyk

5 nununx

41 nununh

0 1 2 3 4123 k

1

knhkx

ny

0 1 2 3 4 5n

6 7

12

3 3 3

5

2

6

1

7

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Convolution Example

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Convolution and LTI systems−Derivation of the convolution sum

x[n] = ∑ x [l] δ[n - l] for l= any integer

= …... x [-2] δ[n + 2] + x [-1] δ[n + 1] + x [0] δ[n] + x [1] δ[n - 1] + x [2] δ[n - 2] +……. x [0] δ[n] x[0] h[n] x [1] δ[n - 1] x[1] h[n - 1] x [2] δ[n - 2] x[2] h[n - 2] x [l] δ[n - l] x[l] h[n - l]

x[n] = ∑ x [l] δ[n - l] y[n] = ∑ x [l] h[n - l]

l

l l

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Properties of LTI Systems−Convolution with an impulse

x[n] * δ [n - n0] = x[n - n0]

−Commutative Property of convolution x[n] * h [n] = h [n] * x[n]

−Associative Property of convolution(x1[n] * x2 [n] )* x3 [n] = x1 [n] * (x2 [n] * x3

[n])

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Assignment # 4−Problems at the end of chapter 5−P-5.2−P-5.3−P-5.4−P-5.6−P-5.10−P-5.12−Not Decided about Deadline Date−Tell you later