Solution Found!
Consider the signal x[ n] depicted in Figure P3.32. This signal is periodic with period
Chapter 3, Problem 3.32(choose chapter or problem)
Consider the signal \(x[ n]\) depicted in Figure P3.32. This signal is periodic with period \(N = 4\). The signal can be expressed in terms of a discrete-time Fourier series as
(P3.32-1): \(x[n]=\sum_{k=0}^{3} a_{k} e^{j k(2 \pi / 4) n}\)
As mentioned in the text, one way to determine the Fourier series coefficients is to treat eq. (P3.32-1) as a set of four linear equations (for \(n = 0, 1, 2, 3\)) in four unknowns (\(a_{0}, a_{1}, a_{2}, \text { and } a_{3}\)).
(a) Write out these four equations explicitly, and solve them directly using any standard technique for solving four equations in four unknowns. (Be sure first to reduce the foregoing complex exponentials to the simplest form.)
(b) Check your answer by calculating the ak directly, using the discrete-time Fourier series analysis equation
\(a_{k}=\frac{1}{4} \sum_{n=0}^{3} x[n] e^{-j k(2 \pi / 4) n}\)
Questions & Answers
QUESTION:
Consider the signal \(x[ n]\) depicted in Figure P3.32. This signal is periodic with period \(N = 4\). The signal can be expressed in terms of a discrete-time Fourier series as
(P3.32-1): \(x[n]=\sum_{k=0}^{3} a_{k} e^{j k(2 \pi / 4) n}\)
As mentioned in the text, one way to determine the Fourier series coefficients is to treat eq. (P3.32-1) as a set of four linear equations (for \(n = 0, 1, 2, 3\)) in four unknowns (\(a_{0}, a_{1}, a_{2}, \text { and } a_{3}\)).
(a) Write out these four equations explicitly, and solve them directly using any standard technique for solving four equations in four unknowns. (Be sure first to reduce the foregoing complex exponentials to the simplest form.)
(b) Check your answer by calculating the ak directly, using the discrete-time Fourier series analysis equation
\(a_{k}=\frac{1}{4} \sum_{n=0}^{3} x[n] e^{-j k(2 \pi / 4) n}\)
ANSWER:Step 1 of 7
Refer figure P3.32 in a textbook and redraw it.
