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Questions tagged [dft]

The Discrete Fourier Transform (DFT) is a mapping between a finite set of discrete points in a (primal) domain (time, space) and the dual frequency domain. DFT requires an input sequence which is discrete, such as a sampling from an analogue audio signal.

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5 votes
3 answers
914 views

Absolute convergence of periodic sinc interpolation

An $N$-periodic complex discrete-time sequence $[x_0, \dots, x_{N-1}]$ can be resampled to an $M$-periodic sequence $[y_0, \dots, y_{M-1}]$ with $M>N$, using sinc interpolation: $$\begin{align}y_m ...
4 votes
3 answers
354 views

Inverse DFT: Is there a valid / intuitive interpretation of results for non-integral timestamps?

I implemented the plain DTF / inverse DFT algorithm in C++ in order to help me understand the method. As a sample input I considered $$ f(x) = \sin \left( \tfrac{\pi}{5} x \right) $$ and collected ...
5 votes
5 answers
2k views

Is it possible to find the FFT of a 1024-point signal by taking 8-input points at a time and calculating the FFT of those 8-points until the end?

Suppose the input signal has 1024 samples, and I want to take a 1024-point FFT. Can I instead take the 8-point FFT of the first 8 input samples then another 8 samples and continue this till the end? ...
0 votes
1 answer
71 views

Generate a Time Series from Power Spectral Density Python

I am trying to generate a time series from a defined PSD function, however i tried to do this in python , with the following steps: Define the Power Spectral Density Define the time parameters Define ...
1 vote
2 answers
147 views

Why is reconstruction of window function from its DFT difficult/noisy when the sidelobes of DFT are higher?

I am able to use the DFT and IDFT formulae to get the DFT of a window and reconstruct it but only if the sidelobes is very low (or the main lobe is narrow) as shown in the image below: Why does this ...
0 votes
1 answer
23 views

Discrete Frequency representation for central frequency/ discretize up-converted signal in time and frequency

I want to analyze a signal after up-conversion in discrete time and frequency, for example: Let's assume a continuous up-converted signal is: $$e^{j2\pi ft} \cdot e^{j2\pi f_c t} = e^{j 2 \pi (f+fc) t}...
0 votes
0 answers
28 views

LED image phase projection

Say you have a goal image G with a phase called phase G. Can you physically construct and project phase G, or is phase not something physical? Like, can you set up an array of LEDs to project the ...
3 votes
1 answer
378 views

Mathematically, why is there a tradeoff between main lobe width and sidelobe level when we apply a window?

This answer says that there is a tradeoff between SLL (sidelobe level) and BW (main lobe width). I am able to verify this and I understand that we cannot have our cake and eat it too, but why is there ...
1 vote
2 answers
86 views

IFFT return complex values in Matlab

I've been experimenting with the frequency sampling method for designing FIR filters. I created a low-pass filter that has a linear transition band. However, when I performed an IFFT on the frequency ...
1 vote
1 answer
45 views

Signal response amplitude depends on the time interval in simulation

I have already tried to look for an answer, but I do not find existing answers satisfactory. I am interested in the absolute value of the response function of a damped oscillator (or any time series). ...
0 votes
2 answers
44 views

Getting the magnitude and phase of a single specific frequency from an audio signal

Assuming I only care to calculate the magnitude and phase of ONE single frequency from a signal, how can I get this information without calculating anything else? For example I want to come up with an ...
0 votes
1 answer
21 views

Constant Q transform where you keep the window size constant (so Q is no longer constant)?

I've been working on implementing a constant Q transform to try and detect musical notes within an audio signal and I came across an issue. When trying to detect low frequencies the constant Q ...
2 votes
4 answers
2k views

Does Zero Padding Distort the Spectrum of a Signal?

It's said to "sample the DTFT", revealing what "DFT fails to see". And I fail to see how this sampling isn't distortion. The "spectrum" aims to provide the sinusoidal ...
2 votes
5 answers
440 views

Why does the frequency sampling method for FIR filter design operate in this manner?

I'm studying FIR filter design and it's time for the frequency sampling method, my teacher said that to use this method you need to follow the following steps: Sample the periodic frequency response ...
0 votes
1 answer
37 views

Are integer multiples of a frequency in an FFT necessarily in phase?

I performed a FFT on a signal and the most prominent frequency at about 0.5 cycles/time only makes up about 6% of the total frequency distribution, which is pretty low for my purposes, so I was ...
0 votes
1 answer
112 views

Unwindowed STFT of sine, closed form solution and insights (sliding FFT)

I seek to calculate, mathematically, the unwindowed Short-Time Fourier Transform of $$ \cos(2\pi f t + \phi) $$ i.e. any arbitrary real-valued sine: any frequency, duration, phase shift, and number of ...
0 votes
1 answer
80 views

Effect of sampling rate and duration on discrete parameters of sine (spectrum)?

The DFT of $$ \cos(2\pi f t + \phi) $$ peaks at $k=\pm f$ if $t = \frac{1}{N}[0, 1, ..., N - 1]$ (for integer $f$, & within Nyquist). What about other $t$? What if we double the sampling rate or ...
2 votes
1 answer
188 views

Finding Discrete Fourier Transform (DFT) for different DFT size

$N$ is an even integer, $x[n]$ is a finite length signal over the interval $n \in [0,N-1]$, and $X[k]$ is the $N$-point DFT of $x[n]$. Analytically find the DFT of sequence below in terms of $X[k]$. ...
10 votes
3 answers
1k views

Why doesnt DFT Padding cause sinc like features

I'm new to the land of DSP so any incorrect terms please let me know. It seems padding the time domain signal can make the magnitude spectrum look 'nicer', the fact it doesn't gain any more useful ...
2 votes
2 answers
194 views

Differences in PSD for windowed vs non-windowed spectra

For a non-windowed spectrum, this article gives this equation for the power spectrum $$\text{PS}(k)=\frac{1}{N^2}|X(k)|^2$$ and this for the power spectrum density $$\text{PSD}(k)=\frac{N}{f_s}\text{...
0 votes
1 answer
82 views

What's Nyquist frequency in DFT?

I'm currently studying digital signal processing at university but I can't figure out what the Nyquist frequency means in the DFT coefficients, I know what the Nyquist frequency is in the sampling ...
3 votes
1 answer
2k views

What are the eigenvalues of the 8 point DFT matrix?

I know that the eigenvalues for 4 point DFT matrix can be found from $F_4^4=I$. Is this also valid for 8, 16 and higher orders? For example with 8 points, will it be $F_8^8=I$ ? If not, how can I ...
2 votes
1 answer
107 views

How signal generation affect its spectrum

Im trying to get dft of hanning window. Im trying different methods of generating the same signal, yet im resulting with such different graph of signal spectrum. ...
0 votes
1 answer
54 views

Non-uniform FFT

I'm looking for a form of a FFT where the samples in the frequency-domain don't represent uniform spaced frequencies. What I would like to get is a frequency-domain with samples that are unevenly ...
0 votes
1 answer
530 views

Applying 2D Sinc Interpolation for Upsampling in the Fourier Domain (DFT / FFT)

Related to The Proper Way to Do Sinc Upsampling (DFT Upsampling) for Uniformly Sampled Discrete Signals with Finite Number of Samples, how can one apply Sinc upsampling in the DFT / FFT domain for a ...
3 votes
2 answers
314 views

Power Spectrum Density and Frequency

if i have some random signals (sampling rate = 10Hz, 0.1s per data) Using python library i transformed it to power spectral density power spectral density forms = f, psd (using mlab.psd) I'm really ...
1 vote
1 answer
69 views

Conjugate symmetric: 3D fourier transform dirmension

I have a real value input 3D tensor with the shape of `(H,W,D)=[8,8,20]', where H, W, and D represent height, width and depth in (z dimension), respectively. When converting to the DFT, what will be ...
2 votes
1 answer
262 views

Proof for the energy correction factor of DFT

I am looking for a mathematical proof for the energy correction factor in conteext of windowed discrete fourier transform. In Spectrum and spectral density estimation by the Discrete Fourier transform ...
2 votes
2 answers
3k views

Effect of cyclic prefix and zero-padding in OFDM

What happens if zeros are added instead of cyclic prefix in OFDM system? How does it affect performance of system?
0 votes
2 answers
96 views

DFT modulus of a sine, closed form solution and insights

A closed form solution to $$ X[k] = \texttt{DFT}\{\cos(2\pi f t + \phi)\} $$ confirmed many known properties of a finite sine's spectrum, also revealed new ones. Can the same be done for $|X|$, or $|X|...
1 vote
2 answers
92 views

DFT Matrix Oversampled In Frequency?

Edit 2: I am trying to replicate results from this paper Compressed Sensing with Coherent and Redundant Dictionaries. On page 3 the "oversampled DFT" is mentioned as an example of an "...
2 votes
1 answer
101 views

How to solve this even symmetry question?

I have my DSP final soon and I have been reviewing some past exams. Here is a question from one of these exams: Let x[n] be a real valued finite duration signal in n $\in [0,N]$. Another signal $x_1[...
4 votes
1 answer
53 views

Show that the similarity between a signal at an analysis frequency with a phase offset with a reference signal is $\frac{N}{2}\text{cos}(\phi)$

I'm reading Brian McFee's excellent book Digital Signals Theory. He presents the DFT as a measure of similarity $S$ between a digital signal $x[n]$ with $N$ samples and a set of reference signals $y_m[...
1 vote
1 answer
185 views

Anyone explain to me this video?

I was watching a video in time 24:48 I would like to know where you got the value $$0.9 (z^2 -1.14z + 0.941)$$ and $$z^2 - 1.0232z + 0.757$$ Does anyone explain how he got those numbers?
0 votes
0 answers
42 views

Is it possible to compute a 2D-FFT if the input size is not a power of 2? [duplicate]

Let us suppose to have an image with size $28\times28$, and that we want to apply a 2D-FFT without any padding operation. Does exist any algorithm which allows to perform the 2D-DFT calculation with ...
1 vote
1 answer
74 views

Expression for DFT of linear 2D ramp

One dimensional solution is in “Expression for discrete fourier transform of linear ramp“ I need two-dimensional for image processing. We have function f(x,y) = $a_1 \cdot x + a_2 \cdot y$, My ...
0 votes
1 answer
90 views

fast spherical filter: interpolation

Here is the minimum working examples of fast spherical filter in c++ The reference paper is Fast Spherical Filter There is a bug in the implementation where the ...
1 vote
0 answers
45 views

Generating new random phases for DFT of 2d uniform distributed noise image changes image distribution

My goal is to produce a new random noise image from two already existing noise images. For that I take the absolute values of one image and the angles (phases) from another image in Fourier Space and ...
4 votes
2 answers
183 views

Why are my frequency bins oscillating?

I am working on a personal project that maps bass notes to colors in real-time. However, I'm encountering some issues with oscillations in my frequency bins. I visualized my frequency bins to ...
2 votes
2 answers
119 views

What is the difference between the DFS (Discrete Fourier Series) and DTFS (Discrete-Time Fourier Series)

I'm looking at two different books written by Oppenheim. In Discrete-Time Signal Processing (source 1) he defines the DFS to be: where, $W_N=e^{-j(2\pi/N)kn}$ , while in Signals and Systems (source 2)...
2 votes
1 answer
227 views

Can time aliasing cause peaks?

For the following I use the terms “time domain signal” and “frequency domain signal” as a Fourier Transform pair. The question is for generalized cases of continuous-time signals that once sampled in ...
2 votes
0 answers
125 views

The Discrete Fourier Transform (DFT) decomposes any signal into four orthogonal signal components

Let $F=\frac{1}{\sqrt{n}}(w^{kl})_{k,l=0}^{n-1}$ be the discrete Fourier matrix of size $n$ where $w=\exp^{-\frac{2\pi i}{n}}$. It is a well-known that $F_n^4 = I_n$ where $I_n$ represents the ...
1 vote
1 answer
193 views

Constraints on choosing the frequency axis when Fourier transforming non-uniformly sampled data?

Does anyone have a reference that specifically discusses choosing the frequency scale for a simple 1D data for non-uniformly sampled time-domain data when performing the discrete Fourier transform. In ...
3 votes
3 answers
2k views

Understanding LPC for Formant Estimation

I went through the Matlab tutorial on Formant Estimation using LPC Coefficients. Though I vaguely understand the details, it's not entirely clear why we need to do this. From http://person2.sol.lu.se/...
0 votes
0 answers
34 views

Why are these excess frequency components in my FFT spectrum? [duplicate]

I am experimenting with digital signal processing and I took the FFT of a sinusoid in Matlab. I got the frequency components that I expected to be present but there are two excess spikes that mirror ...
1 vote
1 answer
129 views

What are the *undesirable* effects of windowing in Fourier space?

My goal is to split a periodic signal into two (or more) signals. The first signal would contain the low-frequency information, and the later signals, the higher-frequency information. These signals ...
0 votes
1 answer
108 views

How are subcarrier symbols modulated into an analog OFDM signal?

I am currently researching about OFDM models (formulae) that show how the individual data symbols (eg., QAM symbols) are converted to OFDM symbols and how the OFDM symbols are turned into an analog ...
1 vote
0 answers
176 views

Is it feasible to implement analog-style spectrum analyzer and sliding DFT as visualizations in music players?

The real-time constant-Q transform are possible using sliding DFT (sDFT) because unlike FFT, sDFT can have arbitrary frequency scale and variable per-bin window size, and of course, the realtime CQT ...
1 vote
2 answers
79 views

do closer peaks Leak more to each other (spectral leakage in FFT)

I am familiar with the spectral leakage problem in FFT due to a rectangle window for a sin signal giving two peaks that leak to each other. My question is will the leakage to the frequency bins near ...
1 vote
1 answer
78 views

How To Calculate Length Of Sequences And A Suitable N?

I have been struggling to understand how to calculate the length of a sequence and also the minimal N to choose in order to avoid aliasing. Most sources tell me to take the (last non-zero value - ...

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