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ENEE 241 FINAL EXAM: SAMPLE PROBLEMS
- Let x(t) = 1 .7 + 3.5 cos(24πt − 2 .1) + 7.9 cos(48πt + 0.8) + 5.4 cos(64πt + 1.1) ,
where t is in seconds.
(i) Determine the fundamental period T 0 of x(t).
(ii) Define (i.e., give the numerical values of) a scalar c and a vector b having the smallest possible length, such that
c = ; b = [ ].’ ; z = c*ifft(b,512) ; x = real(z) ;
generates a vector x consisting of 512 samples of x(t) uniformly spaced in [0, T 0 ).
(iii) Suppose that x(t) is the input to an ideal lowpass filter with unit gain, zero delay and cutoff frequency fc = 15 Hz. Write an equation for the output signal y(t).
- All signals shown below are periodic and have complex Fourier series expansions of the form
s(t) =
∑^ ∞
k=−∞
SkejkΩ^0 t^ ,
where Ω 0 is the fundamental angular frequency.
r ( t ) 1
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 t
... ...
y ( t ) 1
-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 t
... ...
x ( t ) 1
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 t
... ...
(i) Assuming that t is in seconds, what is the value of Ω 0?
(ii) Evaluate R 0.
(iii) Express each Xk in terms of Rk’s.
(iv) Express each Yk in terms of Rk’s. Simplify your answer as much as possible.
- All signals shown below are periodic and have complex Fourier series expansions of the form
s(t) =
∑^ ∞
k=−∞
SkejkΩ^0 t^ ,
where Ω 0 is the fundamental angular frequency. (The curved segments of y(t) are sinusoidal.)
-10 -8 -6 -4 -2 0 2 4 6 8 10 t
y ( t ) 1
... ...
-10 -8 -6 -4 -2 2 4 6 8 10 t
x ( t ) 1
... ...
-8 -4 -4α 0 4 α 4 8 t
r ( t ) 1
... ...
(i) Using the fact that Rk = sin(kαπ)/(kπ), obtain an expression for each Xk. You need not simplify your answer.
(ii) Express each Yk in terms of Xk’s.
- A FIR filter has impulse response
h[n] = b 0 δ[n] + b 1 δ[n − 1] + b 2 δ[n − 2] + b 3 δ[n − 3] + b 4 δ[n − 4]
Its magnitude (amplitude) response is given by
|H(ejω)| = | cos 2ω − 2 cos ω|
and its phase response is piecewise linear.
(i) Assuming that b 0 > 0, determine the values of b 0 ,... , b 4.
(ii) Determine the response of the filter to the input
x[n] =
)n , n ∈ Z
(iii) The phase response ∠H(ejω) of the filter has exactly one discontinuity (jump) in the frequency interval [0, π]. At what value of ω does the discontinuity occur? Solve for ω algebraically (not graphi- cally), using the identity cos 2ω = 2 cos^2 ω − 1.
- Two FIR filters whose impulse response sequences h(1)^ and h(2)^ are given by
h(1)[n] = δ[n] + δ[n − 1] + δ[n − 2] + δ[n − 3] + δ[n − 4]
and h(2)[n] = δ[n] − δ[n − 1]
are connected in series (cascade) to form a single filter with impulse response h.
(i) Using z-transforms or convolution, determine h[n] for every n.
(ii) I claim that if the input to the cascade is a periodic sequence x of period N = 5, then the output sequence y is constant in value, i.e., y[n] = c for all n ∈ Z; and that, furthermore, the value c is the same for all periodic sequences of period N = 5.
Is my claim correct? If it is correct, what is the value of c? Explain.
- Consider the filter with impulse response given by
h[n] =
1 , 0 ≤ n ≤ 8 0 , otherwise
(i) Determine the response of the filter to the input
x(1)[n] = cos
( (^) nπ 2
, n ∈ Z
(ii) Determine the output of the filter to the input
x(2)[n] = cos
( (^) nπ 2
, n ≥ 0
- Consider the following MATLAB code:
h = [ 3 -1 0 1 -3 ].’ ; s = [ 1 -4 2 5 ].’ ; H = fft(h,8); S = fft(s,8); C = H.*S; c = ifft(C);
(i) Determine the vector c.
(ii) Without performing a convolution, determine the vector y obtained by the following code:
h = [ 3 -1 0 1 -3 ].’ ; x = [ 1 -4 2 5 0 0 3 -12 6 15].’ ; H = fft(h,14); X = fft(x,14); Y = H.*X; y = ifft(Y);
