0
votes
1answer
18 views

The bandwidth of the signal $x(t)$.

The bandwidth (B) of the signal $x(t)$ is the range of frequencies (measured on the positive semi-axis) in which $X(\omega)$ takes values ​​different from $0$. Very often $X(\omega)$ is different from ...
2
votes
3answers
168 views

Any good introductory book/tutorial on Fourier Transform (up to FFT) with plenty of exercises and solutions?

I wonder what could be a good book to start learning in depth all aspects of the Fourier transform up to the FFT algorithm, and beyond. I am going to dedicate quite some time on the subject, so I ...
0
votes
1answer
34 views

Question about the frequency domain and the fourier transform

if you have a signal say x(t) in continuous time and you transform it using the Fourier transform for continuous time you get X(w) which is the frequency domain representation of this signal x(t). ...
0
votes
1answer
39 views

Amplitude Spectrum, Nyquist Frequency, mixed/min/max wavelets

The problem is here. Now I know the definition of mixed/max/min phase wavelets, whether the roots lie within the unit circle or not. Starting from n = 1, let $$ x_t = ( 5, 6) $$ $$ X(z) = 5 + 6z $$ ...
0
votes
2answers
55 views

Fourier transform of 1 cycle of sine wave

Consider the signal: $\begin{align*} f(t) &= \sin(\omega t) \tag{$0 \leq t \leq 2\pi/\omega$}\\ &= 0 \tag{elsewhere} \end{align*}$ How to compute the Fourier transform of $f(t)$? I ...
0
votes
1answer
43 views

2-D Fourier Transform of complex exponential with 2-D quadratic phase

I've been looking around to see if there is either an exact transform pair or an approximation to either of the following but have not been able to find anything: $$ \mathcal{F}_{xy}\left( ...
0
votes
1answer
44 views

Multiple Characteristic Function and the Dirac Comb

Given the impulse train(Dirac comb): $$\Delta_T(t)=\sum_{k\in\mathbb{Z}}\delta(t-kT)$$ where $T$ is the signal period, $\delta(t)$ is the Dirac delta function and $\mathbb{Z}$ is the set of integers ...
1
vote
1answer
38 views

CT Fourier Transform

I need to find the Fourier Transform of the given signal below; $$ x(t) = \frac{\sin(\pi t)}{\pi t} \frac{\sin(2\pi t)}{\pi t}.$$ I know that if $ x(t) = \frac{\sin(Wt)}{\pi t} $ , then $ X(w) = ...
0
votes
3answers
357 views

Fourier Series coefficients/Trigonometric functions

I need some help about finding the Fourier Series coefficient of the given signal; $$ x(t) = \sin(10\pi t + \frac {\pi}{6} ) $$ I know that, $$ a_{k} = \frac{1}{T}\int_{0}^{T} x(t)e^{-jkw_{0}t}dt $$ ...
0
votes
2answers
107 views

Fourier Series Coefficient of a given signal

$$ {\rm x}\left(t\right) = \sum_{k = -\infty}^{\infty}\left[\delta\left(t-\dfrac{k}{3}\right) + \delta\left(t-\dfrac{2k}{3}\right)\right] $$ I need to find the Fourier series coefficient of x(t). I ...
3
votes
0answers
55 views

Recovery of Bandlimited Signals

Let $\Omega > 0$ and denote by $\mathcal{B}_\Omega$ the subspace of $L^2(\Bbb R)$ consisting of signals that are bandlimited to $(-\Omega, \Omega)$. Denote $\mathcal{L}_{\Omega} : L^2(\Bbb R) ...
0
votes
0answers
151 views

Q: Calculating Fourier Coefficients and Inverse Fourier Transform

Let $\Omega >0$ and $x \in \mathcal{B}_{\Omega/2}$ is continuous. Define $\hat{y}(\omega) = \sum_{n \in \Bbb Z} \hat{x}(\omega - n\Omega)$. If $\hat{y}$ is expressed as \begin{equation} ...
0
votes
1answer
52 views

Periodic Fuctions - Signals -

If, in the periods, the two half's signal periodic have the same form and opposite phases, the periodic signal has symmetry of half wave. If the periodic signal $g(t)$, of period $T_0$, satisfy the ...
1
vote
0answers
220 views

How can I find the compact trigonometric Fourier series from these signals?

I've been stuck on this for a while, but how exactly would I go about calculating the compact trigonometric Fourier series for both of these signals? I have a general formula down for it, but I just ...
1
vote
0answers
46 views

Discrete time fourier transform of partial sum

I came across the following property of the DTFT: $ \mathcal{F} \Bigg(\sum_{m=- \infty}^{n}x[m]\Bigg) = \frac{1}{1- e^{-j \omega}} X(e^{-j \omega}) + \pi X(e^{-j0}) \sum_{m= ...
2
votes
1answer
493 views

Relationship between DFT index values, frequency in a Fourier series and Hz.

I have a sound file recorded at 44.1 K samples per sec, and some FFT and IFFT algorithms. The sound file is a vector with about $ 2^{17} $ elements. My objective is to find which of the index values ...
0
votes
0answers
328 views

Fourier Series Coefficients for Signals

The question is: We specify the fourier series coefficients of a continuous-time signal that is periodic with period 4. Determine the signal x(t). $a_k=\begin{cases} 0, & k=0\\ ...
0
votes
1answer
114 views

Fourier Series with Signals

So the question is: Determine the fourier series representations for the following signal: Here the formula for the fourier series $$C_k=\frac{1}{T}\int_T \! x(t)e^\frac{-j2\pi kt}{T} \, \mathrm{d} ...
1
vote
2answers
869 views

How can I apply a low pass filter to a Fourier Series?

I have a square wave signal and its Fourier Series. This signal pass through an ideal low pass filter,H(f), which has a cutoff frequency of 4KHz. What is the resulting baseband bandwidth of the ...
1
vote
1answer
670 views

Finding Fourier series with function not centered at the origin

I am trying to find both Fourier cosine and sine series which represent the function F(t) in the interval $(0, \pi)$ where $F(t)=\begin{cases} \frac{\pi}{2} & \ \ 0<t< \frac{\pi}{2}\\ 0 ...
1
vote
1answer
593 views

How can I increase/decrease (frequency/pitch) and phase using fft/ifft

How can I increase/decrease (frequency/pitch) and phase using fft/ifft I think I have the basic code but I’m not sure what to do next PS: It's done in Octave/matlab code Example I have a signal that ...
0
votes
1answer
430 views

Aliasing in DFT: mathematical expression

The Fourier transform of a sinc function is the top hat function. So, if $\{y_k\};\ k\in\{0,1,...,n-1\}$ are samples of the sinc function, sampled $T$ apart, the discrete Fourier transform is ...