The discrete-time Fourier series representation of a signal x[n] with period N is written as \(\rm x[n] = \sum_{k = 0}^{N - 1} a_k e^{j(2kn\pi/N)}\). A discrete-time periodic signal with period N = 3, has the non-zero Fourier series coefficients: a- 3 = 2 and a4 = 1. The signal is
\(\rm 1 + 2e^{ \left( j \frac{2\pi}{6} n \right) }\cos \left( \frac{2 \pi}{6} n\right) \)
DTFS coefficients repeat modulo N = 3: a0 = a−3 = 2, a1 = a4 = 1 and a2 = 0. Thus \(x[n]=2+e^{j2\pi n/3}=1+2e^{j\pi n/3}\cos(\pi n/3)\), the second expression.
Important Questions from Fourier Series
The Fourier series representation of a square wave is shown in the figure below. The fluctuations seen near $x = \pm 1$ are named after which one of the following scientists?
The Fourier series expansion of the function $f(x) = |\cos(x)|$ in the interval $(-\pi, \pi)$ is $\alpha + \beta \left[ \frac{1}{3}\cos(2x) - \frac{1}{15}\cos(4x) + \dots \right]$.
The values of $\alpha$ and $\beta$ respectively are ______.
Periodic function $f(x)$ is given below. $$f(x) = \begin{cases} -1, & \text{when } -\pi < x < 0 \\ 1, & \text{when } 0 < x < \pi \end{cases} ; f(x + 2\pi) = f(x)$$ The CORRECT option representing the Fourier series expansion of $f(x)$ is:
A time-limited waveform $g(x)$ is specified as follows: $$g(x) = \begin{cases} -k, & -\pi < x \le 0 \\ +k, & 0 < x \le \pi \\ 0, & \text{otherwise} \end{cases}$$ A new waveform $f(x)$ is constructed from $g(x)$ as follows: $$f(x) = \sum_{m=-\infty}^{\infty} g(x + 2\pi m), \quad \text{for all } x \in \mathbb{R}$$ The sum of the coefficients of the third harmonics of the sine and cosine terms in the trigonometric Fourier series expansion of $f(x)$ is $\frac{2}{3\pi}$.
What is the value of $k$?
The effective value of the current for the Sawtooth waveform as shown is:
