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Chapter 3: Problem 21

A continuous-time periodic signal \(x(t)\) is real valued and has a fundamental period \(T=8 .\) The nonzero Fourier series coefficients for \(x(t)\) are specified as \(a_{1}=a_{-1}^{*}=j, a_{5}=a_{-5}=2\). Express \(x(t)\) in the form \(x(t)=\sum_{k=0}^{\infty} A_{k} \cos \left(w_{k} t+\phi_{k}\right)\).

Short Answer

Expert verified
The signal is given by \( x(t) = 2\cos\left( \frac{\pi}{4} t + \frac{\pi}{2} \right) + 4\cos\left( \frac{5\pi}{4} t \right) \).

Step by step solution

01

Understanding Fourier Series Representation

A Fourier series for a periodic signal can be expressed as a sum of complex exponentials: \[ x(t) = \sum_{k=-\infty}^{\infty} a_k e^{j \frac{2\pi}{T} k t} \]where \( a_k \) are the Fourier coefficients and \( T \) is the fundamental period. The task requires converting this expression into a sum of cosine terms.
02

Identifying Given Coefficients and Properties

We are given specific Fourier coefficients: \( a_1 = j \), \( a_{-1} = -j \) (since complex conjugate of \( j \) is \( -j \)), \( a_5 = 2 \), and \( a_{-5} = 2 \). Given the property that these coefficients are conjugates, it implies that the terms involve sinusoidal components.
03

Transforming to Cosine Form for Harmonics

The cosine form requires conversion from complex exponential to cosine representation. Using the formula:\[ x(t) = \sum_{k=1}^{\infty} |a_k| \cdot 2 \cos\left( \frac{2\pi}{T} kt + \arg(a_k) \right) \] First, the fundamental period is \( T=8 \), leading to \( \omega_k = \frac{2\pi}{T}k = \frac{\pi}{4}k \).
04

Calculating Cosine Components for k=1

For \( k=1 \):- Amplitude: \( |a_1| = |j| = 1 \). Thus, the term is \( 2 \cdot 1 = 2 \)- Phase: \( \arg(j) = \frac{\pi}{2} \) (since \( j = e^{j\frac{\pi}{2}} \)).Therefore, \[ 2\cos\left(\frac{\pi}{4}t + \frac{\pi}{2}\right) \].
05

Calculating Cosine Components for k=5

For \( k=5 \):- Amplitude: \( |a_5| = 2 \). Thus, the term is \( 2 \cdot 2 = 4 \).- Phase: \( \arg(2) = 0 \) (since 2 is real and positive).Therefore, \[ 4\cos\left(\frac{5\pi}{4}t \right) \].
06

Expressing Combined Signal

Combine the cosine terms from Steps 4 and 5 to express the signal \( x(t) \) as:\[ x(t) = 2\cos\left( \frac{\pi}{4} t + \frac{\pi}{2} \right) + 4\cos\left( \frac{5\pi}{4} t \right) \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Continuous-Time Signals
Continuous-time signals play a crucial role in understanding various phenomena in the real world. In the context of electrical engineering and signal processing, these signals are defined over a continuous range of time. Unlike discrete-time signals, which exist only at discrete time intervals, continuous-time signals are defined for every value of time, allowing them to better model real-world processes such as audio and electromagnetic waves.

A continuous-time signal can be represented using various mathematical and graphical techniques to analyze how it behaves. This helps in the design of systems that need to process these signals, like communication systems. Signal properties like amplitude, phase, and frequency can vary continuously over time, making their analysis distinctive. Handling them effectively requires understanding their mathematical representation through techniques like the Fourier Series.
Periodic Signals
A signal is said to be periodic if it repeats itself at regular intervals over time. For continuous-time signals, being periodic means that the signal remains unchanged every period, noted as \( T \). This property makes it easier to analyze because its behavior over one period can represent the behavior over all time.

In mathematical terms, a signal \( x(t) \) is periodic with period \( T \) if:
  • \( x(t + T) = x(t) \) for all time \( t \).
Periodic signals can be sine waves, square waves, or any wave-like patterns that recur predictably. The Fourier series helps express periodic signals using a sum of sines and cosines, which are themselves periodic. The key insight is that even complex periodic signals can be reconstructed by adding together multiple simple cosine waves of different frequencies and amplitudes.
Cosine Representation
The cosine representation of a signal, particularly within the Fourier Series, allows us to describe the signal as a sum of cosine functions. Cosines are especially useful since they are naturally occurring waveforms in numerous applications.

A general form for a cosine representation of a signal \( x(t) \) using a Fourier series is:
  • \( x(t) = \sum_{k=0}^{\infty} A_{k} \cos \left(\omega_{k} t+\phi_{k}\right) \)
where:
  • \( A_{k} \) is the amplitude of the \( k^{th} \) harmonic.
  • \( \omega_{k} = \frac{2\pi}{T}k \) represents the angular frequency, with \( T \) as the fundamental period.
  • \( \phi_{k} \) represents the phase shift of the \( k^{th} \) component.
The transformation from complex exponential form to cosine form involves capturing both amplitude and phase from the Fourier coefficients. This transformation reveals how the components of different frequencies and phases combine, thereby enabling a complete reconstruction of the original signal. Understanding the cosine representation gives insight into the harmonic content and time-domain behavior of signals.

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Most popular questions from this chapter

Consider three continuous-time periodic signals whose Fourier series representations are as follows: $$x_{1}(t)=\sum_{k=0}^{100}\left(\frac{1}{2}\right)^{k} e^{j k \frac{2 \pi}{50}}$$, $$x_{2}(t)=\sum_{k=-100}^{100} \cos (k \pi) e^{j k \frac{2 \pi}{50} t}$$, $$x_{3}(t)=\sum_{k=-100}^{100} j \sin \left(\frac{k \pi}{2}\right) e^{j k \frac{2 \pi}{50} t}$$. Use Forurier series properties to help answer the following questions: (a) Which of the three signals is/are real valued? (b) Which of the three signals is/are even?

One technique for building a dc power supply is to take an ac signal and full- wave rectify it. That is, we put the ac signal \(x(t)\) through a system that produces \(y(t)=\) \(|x(t)|\) as its output. (a) Sketch the input and output wave forms if \(x(t)=\cos t .\) What are the fundamental periods of the input and output? (b) If \(x(t)=\cos t,\) determine the coefficients of the Fourier series for the output \(y(t)\). (c) What is the amplitude of the de component of the input signal? What is the amplitude of the de component of the output signal?

Consider a continuous-time LTI system \(S\) whose frequency response is \(H(j \omega)=\left\\{\begin{array}{ll}1, & \\{\omega | \geq 250 \\ 0 . & \text { otherwise }\end{array}\right.\). When the input to this system is a signal \(x\) ( \(t\) ) with fundamental period \(T=\pi / 7\) and Fourier series coefficients \(a_{k},\) it is found that the output \(y(t)\) is identical to \(x(t)\). For what values of \(k\) is it guaranteed that \(a_{k}=0 ?\)

Consider a continuous-time ideal lowpass filter \(S\) whose frequency response is $$H(j \omega)=\left\\{\begin{array}{ll}1, & |\omega| \leq 100 \\\0, & |\omega|>100\end{array}\right.$$. When the input to this filter is a signal \(x(t)\) with fundamental period \(T=\pi / 6\) and Fourier series coefficients \(a_{k},\) it is found that \(x(t) \stackrel{s}{\longrightarrow} y(t)=x(t)\). For what values of \(k\) is it guaranteed that \(a_{k}=0 ?\)

When the impulse train \(x[n]=\sum_{k=-\infty}^{\infty} \delta[n-4 k]\) is the input to a particular LTI system with frequency response \(H\left(e^{j \omega}\right),\) the output of the system is found to be \(y[n]=\cos \left(\frac{5 \pi}{2} n+\frac{\pi}{4}\right)\). Determine the values of \(H\left(e^{j k \pi / 2}\right)\) for \(k=0,1,2,\) and \(3\).
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