Filters: Practice Problems

In answering the questions below, please consider the unit sample response and frequency response of two filters, H₁ and H₂, plotted below.

Note: the only nonzero values of unit sample response for H₁ are : h₁[0] = 1, h₁[1]=0, h₁[2]=1.

Note, the only nonzero values of unit sample response for H₂ are : h₂[0] = 1, h₂[1]=-sqrt(3), h₂[2]=1.

In answering the several parts of this review question consider four linear time-invariant systems, denoted A, B, C, and D, each characterized by the magnitude of its frequency response, |H_A(e^jΩ)|, |H_B(e^jΩ)|, |H_C(e^jΩ})|, and |H_D(e^jΩ)| respectively, as given in the plots below. This is a review problem, not an actual exam question, so similar concepts are tested multiple times to give you practice

1. Which frequency response (A, B, C or D) corresponds to a unit sample response given by

   h[n] = α δ[n] - h₁[n]

   and what is the numerical value of |α|? Must be H_B as that is the only frequency response that has the same values at 0 and ±π and extremes at ±&pi/2.

   |α|=2 as H₁(e^j0)=2 but H_B(e^j0)=0.

2. Which frequency response (A, B, C or D) corresponds to a unit sample response given by

   h[n] = Σ_mh₁[m]h₂[n-m] for m = 0 to n

   and what are the numerical values of h[2], h[3] and H(e^j0)? Must be H_A since H₁(e^jΩ)H₂(e^jΩ) = H(e^jΩ), and therefore |H(e^jΩ)|=0 whenever H₁(e^jΩ)=0 or H₂(e^jΩ)=0.

   H(e^j0) = H₁(e^j0)H₂(e^j0) = 2(2 - sqrt(3)) = 4 - 2sqrt(3)

   h[n] = [1,0,1]*[1,-sqrt(3),1] so h[2] = 2 and h[3] = -sqrt(3).

3. Which frequency response (A, B, C or D) corresponds to a unit sample response given by

   h[n] = α δ[n] - Σ_mh₁[m]h₂[n-m] for m = 0 to n

   and what is the numerical value of |α|? Since H_A is the frequency response for H₁*H₂, H_D must be the solution. It's the only frequency response with enough wiggles.

   Since |H_A(e^jπ)| = |H₁(e^jπ)||H₂(e^jπ)|, |α| = 2(2 + sqrt(3)) = 4 + 2sqrt(3).

4. Which frequency response (A, B, C or D) corresponds to a unit sample response given by

   h[n] = α δ[n] - h₂[n]

   and what is the numerical value of |α|? Must be H_C by elimination but also because none of the other frequency repsonse could be generated by a single magnitude shift of H₂(e^jΩ).

   H_C(e^j0) = 0 so |α| = |H₂(e^j0)| = 2 - sqrt(3).

5. Suppose the input to each of the above four systems is

   x[n]=0 for n < 0 and
   x[n] = cos(nπ/6) + cos(nπ/2) + 1.0 for n &ge 0

   Which system (A, B, C or D) produced an output, y[n] below, and what is the value of y[n] for n > 10?

   Must be H_A, as |H_A(e^jΩ)| = 0 for Ω=±π/6 and Ω=±π/2.

   y[n] for n > 10 = |H_A(e^j0)|*1 = 4 - 2sqrt(3)

6. Suppose the input to each of the above four systems is

   x[n]=0 for n < 0 and
   x[n] = cos(nπ/6) + cos(nπ/2) + 1.0 for n &ge 0

   Which system (H₁ or H₂) produced an output, y[n] below, and what is the value of y[22]?

   Must be H₁ since the H₂ system would eliminate cos(π/6), and since the output will eventually be a cosine offset by H₁(e^j0)*1.

   y[22] = 2.