Finding Filter Coefficients

Algebra Level pending

A digital filter has an input x x and an output y y . The filter is implemented as shown below:

y k = a x k + b x k − 1 + c x k − 2 + d x k − 3 \large{y_k = a\,x_k + b \,x_{k-1} + c \,x_{k-2} + d \,x_{k-3}}

Some input / output data from the filter is given below:

What is a 2 + b 2 + c 2 + d 2 a^2 + b^2 + c^2 + d^2 ?

Note: x k x_k denotes the most recent input to the filter, x k − 1 x_{k-1} denotes the second-most recent input to the filter, etc.


The answer is 63.

Steven Chase by Steven Chase , 37, USA

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1 solution

(1)

y 4 = a x 4 + b x 3 + c x 2 + d x 1 y_{4} = ax_{4} + bx_{3} + cx_{2} + dx_{1}

2 = a ( 1 ) + b ( 0 ) + c ( 0 ) + d ( 0 ) 2 = a(1) + b(0) + c(0) + d(0)

a = 2 a = 2

(2)

y 5 = a x 5 + b x 4 + c x 3 + d x 2 y_{5} = ax_{5} + bx_{4} + cx_{3} + dx_{2}

3 = a ( 1 ) + b ( 1 ) + c ( 0 ) + d ( 0 ) 3 = a(1) + b(1) + c(0) + d(0)

3 = a + b 3 = a + b

Substituting the value of a, we get

b = 1 b = 1

(3)

y 6 = a x 6 + b x 5 + c x 4 + d x 3 y_{6} = ax_{6} + bx_{5} + cx_{4} + dx_{3}

− 2 = a ( 0 ) + b ( 1 ) + c ( 1 ) + d ( 0 ) -2 = a(0) + b(1) + c(1) + d(0)

\(-2 = b + c)

Substituting the value of b, we get

\(c = -3\)

(4)

y 7 = a x 7 + b x 6 + c x 5 + d x 4 y_{7} = ax_{7} + bx_{6} + cx_{5} + dx_{4}

4 = a ( 0 ) + b ( 0 ) + c ( 1 ) + d ( 1 ) 4 = a(0) + b(0) + c(1) + d(1)

4 = c + d 4 = c + d

Substituting the value of c, we get

d = 7 d = 7

Therefore,

a 2 + b 2 + c 2 + d 2 a^{2} + b^{2} + c^{2} + d^{2}

= ( 2 ) 2 + ( 1 ) 2 + ( − 3 ) 2 + ( 7 ) 2 = (2)^{2} + (1)^{2} + (-3)^{2} + (7)^{2}

= 63 = 63

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