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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2 answers
162 views

Understanding the result of the fft algorithm

Understanding the result of the fft algorithm. I need help understanding the FFT calculation results. Recently, I have been interested in signal analysis, so I have created and understood fft ...
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0 answers
58 views

Calculating the DFT of the image after using a filter

Studying for my finals in Image Processing course. Trying to solve the following question: Let $h$ be a filter that replaces each pixel value with the average of it's 8 neighbors. Let $f$ be a ...
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1 answer
138 views

Fourier Transform of an 2D image and associated units

I have the following rather simple problem and unfortunately I am not getting forward. Imagine a simple 2D image with pixels and a unique value for each pixel of the image. For example, let the image ...
6 votes
7 answers
1k views

Why is frequency resolution dependent on the number of samples? (need for intuition)

I know the DFT, I agree with the formula and everything, but I don't get the intuition on the link between frequency resolution and number of samples. Like, why would I get a higher frequency ...
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1 answer
117 views

Why are the negative frequencies of the DFT symmetrically reflected at the nyquist to the positive frequencies?

I was playing around with plotting DFT and realized that the negative frequencies are symmetric to the positive frequencies reflected at the nyquist. Plot shown for the signal $f(x) = \cos(\frac{\pi}{...
1 vote
4 answers
128 views

Why is the nyquist frequency at $\frac{N}{2}$ (or $\lfloor \frac{N}{2}\rfloor$) for the DFT and what is the value for $X[k_{N/2-1}]$

For the definition of the DFT we have $X[k] = \sum\limits_{n=0}^{N-1}x[n]\exp(- \frac{2 \pi i \cdot n}{N} k)$ Let's say for simplification that $N$ is even. Then $k_{N/2-1} = \frac{N}{2}$ is ...
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1 answer
78 views

How to calculate the DFT for this sum of cosine's in the form $\sum A_i \cos(\omega_i n + \phi_i)$ for fixed $N$

I am stuck trying to calculate the DFT for a given $N$ Given the signal $x[n] = \cos(\frac{\pi}{2}n - \frac{\pi}{2})+2 \cos(\pi n + \frac{\pi}{2})$ and $N = 4$ I tried to calculate the DFT $X[k] = \...
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1 answer
39 views

DFT algorithm output meanning [duplicate]

The result of calculating the amplitude of the audio through the DFT algorithm above is It came out as below. ...
1 vote
1 answer
597 views

Why is the arithmetic mean the same as the DC component of its fourier transform?

When we define $$\overline{\left|x\right|} = \frac1T\int_0^T x(t) dt$$ as the arithmetic mean of a signal we can see that it is the same as its dc component in the fourier transform. Why is this the ...
4 votes
2 answers
97 views

"DFT". Understanding the formula $e^{-i2\pi k}$ $k$ is a real number

I am studying about fast Fourier transform. Assuming that $x_0$, $x_1, \ldots, x_{n-1}$ are complex numbers, the DFT is defined as follows. $$f_j = \sum\limits_{k=0}^{n-1} x_k e^{-\frac{2\pi i}{n}jk},\...
2 votes
1 answer
314 views

How to interpret a 1D-DFT of an image with a sinus grating/gradient compared to its 2D-DFT outcome?

For getting a better and more intuitive understanding on how the 2D-DFT works I was playing around with sinus gratings in grayscale. I tried to compute the 1D-FFT first and compare it with the 2D-DFT ...
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Why does the 2D-DFT of a sinus gradient not show energy along the diagonal straigh lines and only vertical/horizontal from the diagonal point?

I have been experimenting a little bit with simple examples of the 2D DFT to get a better sense for it's interpretation. For this purpose I have been using sinus gratings with the following code: <...
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1 answer
120 views

Scaling of FFT2 magnitude in image-processing

I got the following code: ...
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54 views

overlap and save method) relation zero padding with double size FIR filter coefficient

i'm doing overlap and save method at frequency domain to do this, i added one block back and forth at filter and signal respectly The problem is my filter is bessel function and that is look like this ...
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1 answer
102 views

iFFT and extracting dw

I have a trivial fourier transform question. I have a correlation function, C(t), with complex components in the time-domain, and dt. I would like it in the frequency domain, C(w), like from ...
5 votes
1 answer
230 views

Applying Convolution in Frequency Domain by Element Wise Multiplication on Time Domain

I'm studying the power spectrum. Right now, I am making a program to try to make sure that the "Fourier transform of the multiplication of some data and window function" and the "...
2 votes
1 answer
103 views

Averaging of a phase flipping signal

Suppose I want to average a signal $s(t)$ which consists of several spectral components without any DC offset. I sample $M$ points in the time domain. I am interested in the power spectrum which I get ...
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1 answer
188 views

SNR of averaging FFT in magnitude

Suppose we record $N$ repetions of a sinusoidal signal with noise (recording $M$ time points). We are interested in the magntiude spectrum. To improve the SNR, I average the signal traces from ...
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2 votes
2 answers
764 views

Relation of zero-padding and frequency resolution

Consider simple rectangular pulse and FFT of it in Python: ...
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Time Scaling Property in DFT

Is there a general Time Scaling property in DFT similar to that of Continuous time Fourier Transform? I couldn't find any when I referred some standard books. This question and its answer don't give ...
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6 votes
1 answer
295 views

The Proper Way to Do Sinc Upsampling (DFT Upsampling) for Uniformly Sampled Discrete Signals with Finite Number of Samples

Given a signal $ \left\{ x [ 0 ], x [ 1 ], ..., x [ N - 1 ] \right\} $ what would be the correct way to upsample it in the frequency domain (Sinc interpolation)? Note: Added as a request by the answer ...
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2 votes
2 answers
196 views

zero-centered and causal zero padding

I have followed the link below to simulate two different zero-padding methods (zero-centered and causal) https://ccrma.stanford.edu/~jos/mdft/Zero_Padding.html Sample code ...
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1 vote
1 answer
179 views

misunderstanding of spectral leakage

I want to understand spectral leakage. I understand that whenever we feed $N$ time-samples of a periodic, continuous, signal into a FFT algorithm we are multiplying in time-domain the true periodic, ...
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2 votes
2 answers
233 views

Misunderstanding of Nyquist sampling theorem and minimum sampling rate

I sample my time-domain (TD) signal using a distance between time-samples of $\delta = (t_{max} - t_{min}) / N_t$, where $N_t$ is the number of samples taken. The sampling rate is $1 / \delta$. I have ...
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How to increase the low frequency resolution of spectrum of music signal

I know normal FFT with a fixed window size will introduce leakage and rectangular window function will give me the best main lobe width. For example, with 44100Hz sample rate, 2048 window size, a 64Hz ...
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1 answer
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DFT explanation for a given signal providing samples per second and total samples

I'm new to the topic of DFT and I need to understand the following question in detail because I'm a bit confused and I need to solve the requirements needed using any programming language so taking ...
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1 vote
2 answers
218 views

Direct calculation of the fft of a rectangular window

For a rectangular window defined as, the frequency spectrum equation and magnitude (or pseudo-magnitude) plot are, However, when I apply Scilab's fft() function or the definition of DFT definition ...
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1 answer
280 views

Using Inverse DFT to reconstruct a sampled sine wave is not perfect?

Start with a 2 Hz signal. The signal is sampled at a 4.167 Hz sample rate. The intent is to reconstruct a sampled signal (top right side) to be identical to the original signal (top left side). ...
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-1 votes
2 answers
60 views

Inserting zeros between the frequencies does not reconstruct the original signal using DFT / Inverse DFT?

I start with a 2 Hz signal. The signal is sampled at a 4.167 Hz sample rate. See https://nyquist.foxping.com/ => "DFT Images 3 zeros.pdf" for images. The intent is to reconstruct the ...
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5 votes
1 answer
183 views

Dealing with the Cyclic Boundary Conditions of Frequency Domain Convolution in Order to Apply Linear Convolution

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Finite Precision DFT looks broken

I calculate the FFT of a sine wave. In Matlab I get the result as I would expect with a nice real and imaginary part. (blue is real, red is imaginary) Now I also calculate a FFT on a FPGA with fixed ...
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2 votes
2 answers
403 views

Is it possible to calculate a 48-point FFT using a 32-point FFT and 16-point FFT?

I have to calculate a 48 point FFT using an N-point FFT library function which only supports lengths that are a power of 2. Is it possible to calculate a 48-point FFT using a 32-point FFT and 16-point ...
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1 answer
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implementing Prime-factor FFT algorithm

I've been trying for days to implement this algorithm to work with size N samples but I can't manage to do it. my goal is to compute FFT for 100 samples, so I need factor 5 and 2, I wrote a simple FFT ...
1 vote
1 answer
168 views

When if an FFT more efficient than Goertzel?

Given a block, $x[n]$, of M samples. Calculating abs(fft(x)).^2 returns the power spectrum of that block through the use of a $M$-point FFT. I can calculate the same using Goertzel's algorithm. At ...
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1 answer
116 views

Non Radix-2 FFT Algorithms

I have 25,200 samples of data. My bandwidth is 12.6KHz and my Fs is 1.26MHz, I want to plot an Amplitude-Frequency Spectrum to display up to 100 different signals, that's on purpose (12,600 * 100 = 1,...
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1 answer
113 views

Why is FFT module for smaller df showing uexpected malfunctioning in Python?

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Inconsistency in DFT in Python

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2 answers
228 views

Frequency Error/CFO effect on Constellation for OFDM

Very often in literature we see that effect of carrier frequency offset is seen as rotating of a constellation points. And explanation is generally that after down converting and sampling the time ...
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1 answer
269 views

FFT of a Time series data

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1 answer
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Fast fourirer transform - Even and odd numbered elements

I'm trying to understand some optimizations on DFT. So in this step, there is a note like the following: The next step involves the mathematical observation that the even-numbered elements can be ...
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2 answers
94 views

"Mirror" look when computing FFT on samples

I'm trying to plot a frequency-magnitude spectrum, I have 2048 samples of a complex sine wave where the imaginary part is always 0, this results a "mirrored" look, so if I have one sine wave ...
1 vote
3 answers
1k views

Interpolation from discrete time fourier transform in python

I have a function that I sample from over one period. I want to use the Fourier Transform to learn the function and then predict unsampled values. Please see the code below: ...
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0 answers
112 views

Which is best method to calculate Sperling’s Ride Index?

I am trying to make a prtotype for measuring Comfort/Ride Index of a testing track road using Sperling’s Ride Index. I looked into a lot of research papers done on this but unable to find the best way ...
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1 answer
49 views

Why multiplying spectral density by 2/N bandwidth results in sinusoidal amplitude?

The image is taken from DSP Guide, where plot [a] is the time domain to synthesize [b] is the frequency domain where the amplitude denotes spectral density and [c] shows the actual magnitude of the ...
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1 vote
1 answer
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Minimum discernible frequency in power spectral analysis

Say I have a signal of length 20s that contains signal from various (unknown) biological sources, e.g. heartbeat (~0.2Hz), respiration (~1Hz), and possibly som very-...
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84 views

Symmetries of discrete analytic signals

Defining "analytic" as $x_a[n]$, where $X_a[k] = \text{DFT}(x_a[n])$, and $$ X_a[k < 0] = 0, \tag{1} \label{1} $$ what time-domain properties, such as symmetry or norm, are guaranteed for ...
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1 answer
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Dirichlet sufficient conditions satisfiability checking

I have 4 signal functions like this: $1.\ e^{-2t}u(t)$ $2.\ e^{-2|t|}$ $3.\ (1-\frac{|t|}{2014})(u(t+2014)-u(t-2014))$ $4.\ u(t)$ I know the answer would be the third option since the other ones have ...
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1 answer
154 views

Fourier transform of an aperiodic discrete-time signal

I have a signal of the form $x^{*}(-n+2)$. To derive the signal's Fourier transform I use The following properties of DFT: $x^{*}(n)=X^{*}(-\omega)$ and $x(n-k) = e^{-j\omega k}X(\omega)$ so the ...
2 votes
1 answer
100 views

Symmetries of analyticity / zero self-correlation

I seek to understand symmetry properties of analytic sequences, without referring to frequency domain: what criteria must a complex sequence $x[n]$ satisfy to be analytic? Framed alternatively, such a ...
1 vote
2 answers
122 views

Parseval's Theorem for Time Domain Data with DC

I have read several times that the variance for time domain data with a zero mean is equal to the integral of the power spectral density divided by N. From wikipedia, the discrete form of Parseval's ...
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