12
votes
Accepted
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 $t$...
- 13.8k
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 ...
- 10.1k
9
votes
Accepted
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 ...
- 29.9k
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 ...
- 40.7k
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 ...
- 80.2k
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 $k\in\mathbb{Z}$
$$
\begin{eqnarray}
&x(t_0+4k\pi) &=\cos(t_0+4k\pi)+\sin(t_0/2+2k\pi)\\
& &=\cos(t_0)+\sin(t_0/2)\\
& ...
- 4,071
6
votes
Accepted
Testing discrete data for periodicity
well, there is always autocorrelation $$ 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 ...
- 16.9k
6
votes
Accepted
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 ...
- 23.7k
6
votes
Accepted
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$-...
- 5,760
6
votes
Accepted
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 ...
- 29.9k
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 ...
- 26.7k
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}$...
- 22.5k
5
votes
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}) $. ...
- 29.9k
5
votes
Accepted
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 ...
- 80.2k
5
votes
Accepted
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 ...
- 40.7k
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+...
- 80.2k
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{ ...
- 40.7k
5
votes
Accepted
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 ...
- 80.2k
5
votes
Accepted
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 ...
- 80.2k
4
votes
Accepted
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 ...
- 40.7k
4
votes
Accepted
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 $...
- 29.9k
4
votes
Accepted
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,
$$Y_k=\sum_{m=0}^Ke^{-j\...
- 80.2k
4
votes
Accepted
Fourier Series Coefficients
You should use the synthesis equation of an impulse train with period $T$ (which is easy to derive):
$$x(t)=\sum_{k=-\infty}^{\infty}\delta(t-kT)=\sum_{k=-\infty}^{\infty}\frac{1}{T}e^{jk\frac{2\pi}...
- 4,105
4
votes
Accepted
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 $X(k)$:
$$\Xi=\dfrac{\sqrt[K]{\prod_{k=0}^{K} X(k)}}{\frac{1}{...
- 10.5k
4
votes
Periodicity of a constant signal!
When you are in doubt, use the limiting approach as an aid in your deductions:
For example, you can consider a constant signal $x_C(t)=1$ as the limit of a periodic sine wave $x_p(t)= \cos(\omega_0 t)...
- 26.7k
4
votes
Accepted
Is Oversampling a Signal Same as Discretizing the Signal?
Oversampling is the case the rate the data is sampled is higher than required by the data Bandwidth and Nyquist Shannon Sampling Theorem.
It has many good properties regarding the processing yet it ...
- 40.7k
4
votes
Accepted
How to Extract a Period of a Periodic Pulsed Signal?
Natalia Molinero Mingorance, Welcome to the DSP community.
What you have is basically a shifted periodic signal.
Why? Because what you have is equivalent (Given many samples) of having a periodic ...
- 40.7k
4
votes
How do I find the fundamental period of the given signal?
If your top equation is really
$$
x(t) = 2\cos\left(\frac 45 \pi t\right)\sin^2\left(\frac{16}{3} t\right)\tag{1}
$$
You gonna have a hard time getting the fundamental period/frequency as the there ...
- 3,272
4
votes
Change in frequency on differentiation
No, in a conventional sense of a "periodic signal" phrase, but, if you permit me to delve into a math subtlety, differentiation can turn an aperiodic waveform to a periodic one:
$$
\frac{\...
- 1,415
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