12 votes
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Periodicity of a constant signal!

As you say, the constant function is periodic. A signal $x(t)$ is said to be periodic with period $p$ or to have a period $p$ if there exists a $p>0$ such that $x(t+p)=x(t)$ for all real numbers $t$...
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10 votes

How can a signal be both periodic and random?

Most realistic signals are both random and periodic. For example, you can modulate a harmonic oscillator with a slow enough random signal that moves its frequency around a $\mu_{f}, \sigma_f$. This ...
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9 votes
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Fourier transform artifacts

The cross pattern is typically a border effect, due to the periodicity induced by the standard implementation and hypotheses behind the Fast Fourier transform, when the image lacks periodicity from ...
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9 votes

Simple and Effective Method to Estimate the Frequency of a Single Sine Signal in White Noise

I assume the model to be: $$ x \left[ n \right] = \sin \left[ 2 \pi \frac{f}{ {f}_{s} } n + \phi \right] + w \left[ n \right] $$ Where $ w \left[ n \right] $ is white noise uncorrelated with the ...
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8 votes

$2\pi$ periodicity of discrete-time Fourier transform

The argument does not work in continuous time. In discrete time the argument is that $$e^{j\omega n}=e^{j(\omega+2\pi)n},\qquad n\in\mathbb{Z}\tag{1}$$ This is true because by definition $n$ is an ...
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8 votes

Is $x(t) = \cos t + \sin\left(\frac{1}{2}t\right)$ a periodic signal?

As for every $t_0\in\mathbb{R}$ and $k\in\mathbb{Z}$ $$ \begin{eqnarray} &x(t_0+4k\pi) &=\cos(t_0+4k\pi)+\sin(t_0/2+2k\pi)\\ & &=\cos(t_0)+\sin(t_0/2)\\ & ...
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6 votes
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Testing discrete data for periodicity

well, there is always autocorrelation $$ R_x(\tau)=\sum x[n] x[n+\tau] $$ or AMDF $$ Q_x(\tau) = \sum |x[n] - x[n+\tau]| $$ or ASDF $$Q_x(\tau) = \sum (x[n] - x[n+\tau])^2 $$ with the latter there is ...
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6 votes
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Why is cos(n/6) aperiodic?

The problem with your reasoning is that $\pi \ne \frac{22}{7}$; $\pi$ is an irrational number. There is no period $N$ for which $x[n] = x[n+N] \ \forall \ n \in \mathbb{Z}$. Hence, the sequence is not ...
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6 votes
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How to calculate the FFT period

You must have understood the notion of digital linear modulation or discrete time vs continuos time (see Chapter 2). Another reference. OFDM can be thought as FDM with sinc pulse whose delay-$T$-...
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6 votes
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How can a signal be both periodic and random?

If you are talking about a given signal as "a deterministic realization of a phenomenon", it can be periodic, but not really random. However, some physical systems are prone to produce randomness ...
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6 votes

Why Zero Padding in the Center of the DFT Interpolates / Upsamples the Signal (Sinc Interpolation / DFT Interpolation / Periodic Interpolation)

What you are experiencing is technically called interpolation by DFT; i.e., interpolating a time-domain sequence $x[n]$ by properly zero filling the middle portion of it's DFT $X[k]$ (and taking the ...
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5 votes

Is $x(t) = \cos t + \sin\left(\frac{1}{2}t\right)$ a periodic signal?

To add a contrarian answer: If your time index, $t$, is an integer, then your signal is not periodic. The definition of periodic is: $x[t]$, $t\in\mathbb{Z}$ is periodic with period $P\in \mathbb{Z}$...
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Is $x(t) = \cos t + \sin\left(\frac{1}{2}t\right)$ a periodic signal?

The double-angle formulae for trigonometric identities tell you that $\cos \left(\frac{2t}{t}\right) = 1 - 2\sin^2(\frac{t}{2})$. You thus ave $x(t) =1 +\sin(\frac{t}{2}) - 2\sin^2(\frac{t}{2}) $. ...
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Is the periodogram squared-magnitude DFT or squared-average DFT?

The periodogram is simply the squared magnitude of the DFT. Since the periodogram is a rather poor estimate of the power density spectrum of a random process there are methods which use averaging of ...
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Detecting Pattern from Signal Data by Gaussian Mixture Model?

If the data is cyclic by its nature the best thing would work using its spectrum. You can easily build a system which checks sub set of data to verify periodic and the once you establish your groups ...
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5 votes

formula derivation for periodic signal power

I'd like to show you a more formal derivation. Note that the first formula for arbitrary (non-periodic) signals could be rewritten as $$P_x=\lim_{M\rightarrow\infty}\frac{1}{(2M+1)N}\sum_{n=-MN}^{(M+...
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5 votes

Fourier Transform of Alternating Periodic Rectangular Pulse

The answer is simple. I will give 3 points to solve it: The Fourier transform is linear. Hence $ \mathcal{ F } \left\{ \alpha f \left( x \right) + \beta g \left( x \right) \right\} = \alpha \mathcal{ ...
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Periodic signals in Continuous and discrete time

A periodic continuous-time signal satisfies $x(t)=x(t+T_0)$ for all $t$. The period $T_0$ doesn't need to be a rational number. A periodic discrete-time signal satisfies $x[n]=x[n+N]$ for all integers ...
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5 votes
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Fourier series of cycloid

The Fourier series of the cycloid can be expressed in terms of the Bessel functions of the first kind: $$J_n(x)=\frac{1}{\pi}\int_0^{\pi}\cos(nt-x\sin t)dt,\qquad n\in\mathbb{Z}\tag{1}$$ Using the ...
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4 votes
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How to Calculate the Period of the Discrete Time Sequence from Taking Its DFT

Sample the signal into MATLAB. Apply the fft function on it. Plot the absolute value of the DFT signal (Its first half of samples). Look for its peak, the index ...
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4 votes
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Power of a periodic sequence

The basic trick is to bound the series above and below. Let us do it on one side, for positive indices. For any $N> 0$, you can write $N=kN_0+r_N$, with $0\le r_N< N_0$. Then if $a_n$ (here $...
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4 votes
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DFT and periodicity

As you have correctly observed, $2N/W$ must be an integer, because the window can only have an integer number of samples. Furthermore, regardless of the upper summation limit, $$Y_k=\sum_{m=0}^Ke^{-j\...
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4 votes
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Fourier Series Coefficients

You should use the synthesis equation of an impulse train with period $T$ (which is easy to derive): $$x(t)=\sum_{k=-\infty}^{\infty}\delta(t-kT)=\sum_{k=-\infty}^{\infty}\frac{1}{T}e^{jk\frac{2\pi}...
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4 votes
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periodicity coefficient

I suggest using Spectral Flatness, aka Wiener Entropy. It is defined as a ratio of geometric and arithmetic mean of the magnitude spectra $X(k)$: $$\Xi=\dfrac{\sqrt[K]{\prod_{k=0}^{K} X(k)}}{\frac{1}{...
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4 votes

Periodicity of a constant signal!

When you are in doubt, use the limiting approach as an aid in your deductions: For example, you can consider a constant signal $x_C(t)=1$ as the limit of a periodic sine wave $x_p(t)= \cos(\omega_0 t)...
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Is Oversampling a Signal Same as Discretizing the Signal?

Oversampling is the case the rate the data is sampled is higher than required by the data Bandwidth and Nyquist Shannon Sampling Theorem. It has many good properties regarding the processing yet it ...
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4 votes
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How to Extract a Period of a Periodic Pulsed Signal?

Natalia Molinero Mingorance, Welcome to the DSP community. What you have is basically a shifted periodic signal. Why? Because what you have is equivalent (Given many samples) of having a periodic ...
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4 votes

How do I find the fundamental period of the given signal?

If your top equation is really $$ x(t) = 2\cos\left(\frac 45 \pi t\right)\sin^2\left(\frac{16}{3} t\right)\tag{1} $$ You gonna have a hard time getting the fundamental period/frequency as the there ...
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4 votes

Change in frequency on differentiation

No, in a conventional sense of a "periodic signal" phrase, but, if you permit me to delve into a math subtlety, differentiation can turn an aperiodic waveform to a periodic one: $$ \frac{\...
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