# 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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### Reconstructing the original signal from its DFT

Hi I am a newbie to signal processing and I am trying to better understand how inverse DFT works under the hood. Consider this signal and its DFT: (Source) For the sake of this post, let's assume ...
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### Discrete Fourier Transform of the Gaussian

Cross-posted from here I encountered the following question in a Digital Image Processing examination: Find the 2D DFT of $\frac{1}{2 \pi \sigma^2} e^{-\frac{(x - x_0)^2 + (y - y_0)^2}{2 \sigma^2}}$ ...
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### Driving bandpass DSM with tone close to full-scale - stability issue?

I am experimenting with Dr. Schreier's DSM toolbox for MATLAB. I stumbled over a strange phenomenon when having a look at a bandpass DSM in demo-script 2. If I increased the amplitude of the test-tone ...
1 vote
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### please help me solve the question which proving X[N-k] is the complex conjugate of X[k] [closed]

Show for the DFT that if all x[n] are real, then X[N - k] is the complex conjugate of X[k] for k > 0. (i.e. if X[k] is a + bi, then X[N - k] is a - bi)
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### How to derivate the DFT of the time-domain samples?

I am currently reading a paper titled 'Channel Estimation and Data Detection for Insufficient Cyclic Prefix MIMO-OFDM' published in IEEE Transactions on Vehicular Technology. As you can see below, the ...
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### 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 $X[k]$. ...
1 vote
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### Is fft always the best choice?

I hope this is the right place to ask this question since it is partially a note which might help others. Until recently, I always used the fft-algorithm ...
1 vote
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### Amplitude scaling of window functions for FFT

I need to perform an FFT on a signal sampled with 20 kHz and a measurement time of about 10 seconds. The signal contains frequencies of up to 2 kHz but I am mainly interested in the bandwidth of 0 to ...
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### High Resolution Spectral Analysis

Inspired by this post and this video I have been playing around with different methods in Matlab to do spectral analysis. In the past I have simpy used digital fourier transforms and played around ...
1 vote
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### Effect of windowing on signals

I have an input signal of frequency 1000Hz. Sampling frequency=16kHz. I compute the FFT on 128 input samples and plot the magnitude spectrum. I am not doing the normalization by 128. My two sided ...
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### 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 ...
1 vote
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### How can the DCT be used for bandlimited interpolation?

I know the discrete cosine transform (DCT) is used for compression, but can anyone give an example of how to use it for bandlimited interpolation? One way might be zero-padding in the DCT domain and ...
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### 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 ...
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I'm facing confusion about the definition of the convolution between two discrete periodic signals. Basically, the definition of convolution between s and ...
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### Convolution error when using DFT for non-periodic functions

I need help understanding why FFT-based computation of convolution on a finite domain, between two non-periodic functions often gets it wrong. Specific questions: (1) why does FFT-based computation ...
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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|... 0 votes 1 answer 63 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 95 views ### Does subtracting a phase from the frequency components of a DFT output result in rotation? I'm currently reading a paper (Page 23 of the PDF) about the application of the Fourier transform to standardize some climatic data to easily compare them. I have the following text : 2.3 ... 2 votes 3 answers 774 views ### Why does spectrum magnitude decay away from DC for positive signals? I would like to understand why the second frequency component usually has the greater magnitude within the range [1, (N/2)], i.e why (I remove the DC component and ... 1 vote 2 answers 105 views ### STFT needs circular shift or not? In some textbooks and websites, circular shift operation is done before doing FFT of windowed data, in the explaination, circular shift is to ensure zero-phase. But in other textbooks and websites(I ... 9 votes 3 answers 1k views ### Why do sinusoids have DFT magnitudes of N / 2 while we typically normalize by N? I'm wondering why evaluating a sinusoid that matches one of the frequencies of the DFT basis functions has a magnitude of$N / 2$. Using this definition of the Discrete Fourier transform, it looks ... 1 vote 3 answers 253 views ### Increasing STFT resolution by repeating the window? Ways to improve STFT resolution? In theory of FT(Fourier transform) and STFT(Short Time Fourier Transform) it is said that "A narrower window gives good ... 0 votes 0 answers 30 views ### non-integer phase estimation after dft(using psf) I'm studying PSF function and the way to get the most accurate phase estimation at the non-integer locations. My professor showed me a slide in lecture. Rotate some integer coordinate in fourier ... 0 votes 0 answers 74 views ### How many samples are needed to compute an DFT? Given a signal$x(t) = \frac4{10}\cos(800πt) + \frac12\cos(820πt) + \frac1{10}\cos(880πt)$and knowing that the sampling frequency is$4000$Hz. How many samples are needed at least to represent the ... 0 votes 1 answer 93 views ### How to produce a causal, linear phase filter kernel with an arbitrary magnitude frequency response I have been reading The Scientist and Engineer's Guide to DSP to learn how to create filter kernels with an arbitrary frequency response (I design the magnitude response by hand). The method proposed ... 0 votes 0 answers 26 views ### How to determine the size of DFT bins to implement a bandpass filter with FFTW on SDR I'm implementing a bandpass filter to isolate the amplitude of a UHF FM signal with a 12.5Khz bandwidth. I'm having difficulty isolating the correct bins and bin spacing, given my DFT to isolate the ... 2 votes 1 answer 89 views ### Custom DFT filter adding odd sound to audio I am attempting to design a custom filter for equalizing an audio track using the DFT. I am newish at this, but my understanding from the DSP Guide is that you can do this with the knowledge that ... 1 vote 1 answer 77 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[... 1 vote
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### Is it possible to consider circular convolution in case of appending the guard interval as a suffix instead of a prefix

In OFDM system, the cyclic prefix is added into the head of time-domain signal to enable the circular convolution with the channel, and then perform one-tap frequency-domain equalization at the ...
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### What is the reason my DFT of a convolution of signals with disjoint spectrums is not zero everywhere?

Assuming the convolution in the time domain produces a signal which spectrum is the product of individual spectrums, when the two convoluted signals are sinusoids with different frequencies, the DFT ...