Questions tagged [fourier-transform]

The Fourier transform is a mathematical operation that decomposes a function into its constituent frequencies, known as a frequency spectrum.

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258 views

Cross-correlation, sharp peak at 0?

First of all, I have to stress that I am not a professional of coding, no more than a professional of signal processing. I am a chemist that happen to be working on a project involving both. So in ...
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1answer
2k views

STFT amplitude normalization, librosa library

So librosa.core.stft returns a complex single sided spectrogram. My question is: What normalization of the amplitude values should I perform afterwards? I believe I have to multiply the amplitude ...
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100 views

Question about time shifting property

I know that time shifting property of discrete-time Fourier Transform states that $x[n]\leftrightarrow X(e^{j\omega})\implies x[n-n_0]\leftrightarrow e^{-j\omega n_0}X(e^{j\omega})$ So given a signal ...
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1answer
176 views

Bandwidth of Information Signal

I have trouble finding the bandwidth of a signal. Say I have an info bearing signal m(t)=sinc(2t/pi). I found the fourier transform of the sinc function and found that the angular frequency was 1/pi. ...
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1answer
64 views

Discretization of Sinusoidal Signals

How do I should sample two continuous sinusoids of $f_1$=1/4 Hz and $f_2$=1/2 Hz at a 1 Hz rate? Can i recover both if i bandlimit after sampling at a 1/2 Hz limit?
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268 views

Should scipy.signal.coherence be 1 for single input and output signals?

I am trying to calculate the coherence between input and output signals. I thought I could work with a single input and a single output time series and calculate the coherence $\gamma^2$ between them. ...
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178 views

Signal squared in time domain

In some publication I have found a formula that for some reason squares signal in time domain. I am analyzing it now. What should be the Fourier transform of a signal that was squared in time domain? ...
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1answer
125 views

CMV: COLA constraint is not necessary

In the literature on the short-time Fourier transform (STFT) it is often stated or implied that the constant overlap-add (COLA) constraint must be satisfied in order for the STFT to be invertible. ...
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313 views

How to calculate a delay between two signals in frequency domain?

Calculation of a delay between two signals is generally done by finding the maximum in the cross-correlation of the two signals (in time domain): $$ \tau_{delay} = \arg\max_\limits{t}(f(t) \star g(t)) ...
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653 views

Hilbert transform of unit step function

How to calculate Hilbert transform, if it exists, of the signals like $u(t)$, $sgn(t)$. What properties should a function satisfy for existence of Hilbert transform. Absolute integrability of a ...
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74 views

How does sampling rate of $x[n]$ relate to sampling rate in frequency domain after DFT transformation?

I've got an analog signal $x(t)$ sampled at frequency $F_s$ to obtain samples: $$ x[n] = x(t) \bigg|_{t=n/F_s} $$ I transform this signal with DFT defined as: $$X[k] = \sum_{n=0}^{N-1}x[n]e^{-i2\pi ...
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Wikipedia equation for DFT seems to be bad?

I was writing a simple fourier transform implementation and looked at the DFT equation on wikipedia for reference, when I noticed that I was doing something differently, and after thinking about it ...
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85 views

Find cepstrum of a function with dirac function

$$x[n] = d[n] + a\cdot d[n − N_p]\ ,\ \ |a| < 1$$ I need the complex and real cepstrum of this function as well as what would happen for $N=N_p/6$ for the DFT approximation $$x_p[n] = \frac{1}{N} ...
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157 views

Uncertainty Relation between time and frequency

Typically, humans can hear sound waves in the frequency range 20 Hz to 20 kHz. If one wants to make digital record of sound such that no audible information is lost, what is the longest time interval ...
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649 views

Why is the Fourier transform valid only for absolutely integrable signals?

Why is the Fourier transform valid only for absolutely integrable signals? For example, why can't we do the Fourier transform of exponential order functions?
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407 views

Amplitude of signal and PSD peak height

So, suppose you have a superposition of cosines of the same magnitude: $\sum_{i}\cos[\omega_i t]$ When you take the Fourier transform, theoretically you'd expect to find $\delta$-like peaks of the ...
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1answer
104 views

Inverse DFT to a real signal when only odd harmonics are present

There is an old paper by L.R. Rabiner, "On the Use of Symmetry in FFT Computation" which describes (among other things) an optimized method to calculate the DFT of a real signal $x_n$, $n=0,\dots, N-1$...
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1answer
6k views

How to find the phase spectrum of a rectangular pulse? (Fourier Transform)

How to find the phase spectrum of a rectangular pulse? The Fourier transform of a rectangular pulse $$ x(t) = \begin{cases} 1, & \text{for $|t| \le \tau /2$ } \\ 0, & \text{otherwise} \end{...
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Understanding Magnitude Spectrum of Images [closed]

I am facing problem in reading Fourier domain of a given image. I don't understand what to interpret from it. For instance, consider this image. Ok, so there is a dot in the middle, with some ...
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Demonstrate that $x(t)$ and $\hat{x}(t)$ are orthogonal

I have to demonstrate that a function $x(t)$ and his Hilbert's transform are orthogonal, it is said: $$\int^{\infty}_{-\infty} x(t) \cdot \hat{x}(t) dt = 0$$ I have tried the exercise using Parseval'...
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Why is the size of the output of the FFT of a signal is same as the size of the input signal?

I am kind of new to the DSP domain. I was trying to get the frequencies associated with a signal by performing FFT over it. I used numpy.fft.fft for this. ...
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183 views

Understanding where the constant $2/N$ comes from in Fourier transformation

I'm implementing Fourier transformation in my analysis and I wanted dig a bit deeper on the reasons why the absolute value of Fourier transformation is usually multiplied by the constant $2/N$ to get ...
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204 views

output shape of STFT

I cant figure out why our output image is 257 * 32 , 32 I know why , but 257 I cant understand ? this is the exact word of this paper wich applied STFT on 3 channel of eeg signal : Short time Fourier ...
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229 views

Applications of Power Spectral Density [closed]

I have a class covering Power Spectral Density but I have no idea why it matters. Could someone provide some examples of its use? Thanks
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196 views

Fourier Transform Units

It is documented that 'one' of the units of the Fourier Transform [of $x(t)$ volt] is volt per Hz. That is $X(\omega)$ components will have units of volt per Hz, where $\omega$ is the angular ...
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51 views

Fast Fourier Transform MATLAB [duplicate]

I dont understand one thing when I use the function fft(x,N) in matlab, where x is the signal which I want to calculate the fourier transform and N is the number of samples. What I dont understant ...
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New to audio processing, why is my data so lopsided when I try to break into frequency ranges?

I have a wav file audio data, I broke it up into 1024-length windows (no overlap), and performed fft on each one. If I visualize this data it actually looks pretty good, but the problem is that the ...
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1answer
91 views

FFT - second and further divides and conquers - need help

​ ​Hello, I would like to ask you for help in understanding Fast Fourier Transform. Most articles about FFT describe a simple DFT example with N=8 number of samples. They divide it on half, to evens ...
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275 views

Why look at power spectral density for stochastic processes?

I have been told that for deterministic signals, it makes sense to look at their respective Fourier transforms/spectra. For stochastic processes on the other hand, I am supposed to work with power ...
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101 views

Characteristic and moment generating function of a random variable interpretation

I have been studying about moments and cumulants of a random variable. Even though the definitions of characteristic and moments generating function are very similar (only the sign in the exponential ...
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2answers
57 views

Fourier Transform Signals - Time Transformations

I was going over some review problems and came across an interesting one. Using the techniques of (linearity, time shifting, and time scaling) what are some approaches I could use to turn the ...
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584 views

Correct magnitude spectra of a cosine DFT?

I've just started my course on DSP and haven't laid my hands on MATLAB yet. I was wondering if the plot of the magnitude spectra was correct for the below shown $x(n)$:
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456 views

Why edge sharpening produces high frequency?

I have a low-resolution image in which the high frequencies are missing. When I apply an edge sharpening filter some of the missed high frequency is recovered. I am wondering why this edge sharpening ...
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1answer
168 views

Wiener Filter Additive Noise Uncorrelated

I have been trying to understand for a while now why a non-causal Wiener filter that has a frequency response of the type H(jω)=S_yx(ω)/S_xx(ω) where ...
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0answers
101 views

When Is a Power of 2 FFT Slower than Smaller Sized Non Power of 2 FFT in MATLAB

Knowing that computing an FFT is faster if the amount of samples is a power of 2 I have always tried to pad the inputs to Matlab's FFT with zeros until the next power of 2 is achieved. Matlab's ...
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1answer
2k views

Impulse response of ideal filters

I am aware that an ideal low-pass filter in both continuous time and discrete time has a $\mathrm{sinc}$ impulse response. What would the impulse response of an ideal high-pass or band-pass filter ...
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2answers
201 views

Simple $\sin(2\pi 1000t)$ Fourier transform in PSpice not behaving as expected

So I have this very basic circuit show below which I am simulating with PSpice. Now, when doing the Fourier transform of $\cos(2\omega1000t)$, I expect to see two impulses(one at -1kHz and one at ...
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2answers
424 views

BIBO Stability and the convergence of the frequency response of a system

It is my understanding that an LTI system is BIBO stable if and only if its impulse response $h(t)$ is absolutely integrable. This also happens to be one of the Dirichlet conditions for the ...
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1answer
274 views

Time scaling of discrete-time sequences and the DTFT

In the second edition of Signals and Systems by Alan Oppenheim, he discusses the DTFT of a "time-expanded" sequence that is effectively a slowed down version of the original sequence and can be ...
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1answer
515 views

Computing the center of mass of a signal using the Fourier transform

I am having difficulty with a homework problem which asks: I have some ideas in mind but I have no clue as to whether they are correct or not. Below is my attempt:
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2answers
433 views

Fourier transform of even/odd parts of a complex signal

Why does Oppenheim state the following properties: \begin{align} \mathcal F\big\{x_e (t) \big\} &= \Re\big\{ X(j\omega) \big\}\\ \mathcal F\big\{x_o (t) \big\} &= j \Im\big\{ X(j\omega) \big\...
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1answer
94 views

Uniqueness of Fourier Series Representation and the Fourier Transform of Periodic Signals

If we are given a signal of the form $$x(t) = \sum_{k = -\infty}^{+\infty} a_k e^{j k \omega_0 t},$$ can we call it a Fourier Series representation of $x(t)$ right away? Suppose we are given the ...
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1answer
639 views

Calculating SPL from pressure signal - Amplitude vs Power method

I have a pressure signal from a Fluent FFowcs-Williams Hawkings acoustics analysis. I converted this pressure signal into the frequency domain in order to get SPL values, using Matlab. I used the ...
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3answers
441 views

Periodicity of the discrete-time Fourier Transform

The DTFT of a sequence $x[n]$ can be written as $$X(e^{j\omega}) = \sum_{n = -\infty}^{\infty} x[n] e^{-j\omega n}.$$ Is the smallest (fundamental) period in frequency of the DTFT always $2\pi$? Or ...
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1answer
1k views

Result of inverse FFT is sometimes shifted in real space

I am using the Numpy fft2, ifft2, and related functions and I am sometimes running into a strange situation where the output ...
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141 views

What is the exact meaning of the output of the Discrete Fourier Transform

I'm fairly new to the subject, but so far my understanding that this would be a transform you could use to go from a discrete set of data, say [1, 0, 1, 2] to a continuous sinusoidal function in the ...
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2answers
102 views

Duality of the continuous-time Fourier transform - derivation and notation

Suppose we have the Fourier transform pair $x(t)$ and $X(\omega)$ such that $$X(\omega) = \int_{-\infty}^{\infty} x(t) e^{-j\omega t} \mathrm{d}t$$ The duality property states that $X(t)$ and $2\pi ...
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1answer
56 views

Is there a name for the procedure of taking the FT over separate consecutive small time-blocks?

Suppose we have a continuous time-interval $I=[a,b]$, and a signal $x \colon I \to \mathbb{R}$. A procedure that is sometimes carried out (e.g. when doing bispectral analysis) is to partition $I$ ...
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2answers
107 views

fourier transform of smart-phone accelerometer in matlab

I'm new to matlab. I want to process my smart-phone accelerometer data in matlab. I know Matlab let's you connect your phone via USB cable to see accelerometer data in realtime. But according to some ...