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Problem 4. (16 POINTS; PART (i) 5 POINTS, PART (ii) 6 POINTS, PART (iii) 5 POINTS) Recall that a family of discrete-time harmonically-related complex exponentials (HRCEs) is: $x_k[n] = e^{j\frac{2\pi k}{N}n}$, $k = 0,..., N - 1$. In this problem, consider the family with fundamental frequency $\omega = \frac{\pi}{4}$. (i) Write the expression for this HRCE signal family. What is the value of $N$? HRCE family: $N = $ (ii) Plot the numbers $x_1[1]$ and $x_3[6]$ on the complex plane diagrams given below. $x_1[1]$ Imaginary $x_3[6]$ Imaginary Real 2 1 Real 2 (iii) Which signal in the family $x_k[n]$ has the highest fundamental frequency? Write the formula for this signal in its most simplified form. $k = $ $x_k[n] = $

          Problem 4. (16 POINTS; PART (i) 5 POINTS, PART (ii) 6 POINTS, PART (iii) 5 POINTS)
Recall that a family of discrete-time harmonically-related complex exponentials (HRCEs) is:
$x_k[n] = e^{j\frac{2\pi k}{N}n}$, $k = 0,..., N - 1$.
In this problem, consider the family with fundamental frequency $\omega = \frac{\pi}{4}$.
(i) Write the expression for this HRCE signal family. What is the value of $N$?
HRCE family:
$N = $
(ii) Plot the numbers $x_1[1]$ and $x_3[6]$ on the complex plane diagrams given below.
$x_1[1]$
Imaginary
$x_3[6]$
Imaginary
Real
2
1
Real
2
(iii) Which signal in the family $x_k[n]$ has the highest fundamental frequency? Write the formula
for this signal in its most simplified form.
$k = $
$x_k[n] = $

Problem 4. (16 POINTS; PART (i) 5 POINTS, PART (ii) 6 POINTS, PART (iii) 5 POINTS)
Recall that a family of discrete-time harmonically-related complex exponentials (HRCEs) is:
xk[n] = e^j(2π k)/(N)n, k = 0,..., N - 1.
In this problem, consider the family with fundamental frequency ω = (π)/(4).
(i) Write the expression for this HRCE signal family. What is the value of N?
HRCE family:
N =
(ii) Plot the numbers x1[1] and x3[6] on the complex plane diagrams given below.
x1[1]
Imaginary
x3[6]
Imaginary
Real
2
1
Real
2
(iii) Which signal in the family xk[n] has the highest fundamental frequency? Write the formula
for this signal in its most simplified form.
k =
xk[n] =

Added by Paul N.

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