23 votes
Accepted

Fourier transform 4 times = original function (from Bracewell book)

I'll use the non-unitary Fourier transform (but this is not important, it's just a preference): $$X(\omega)=\int_{-\infty}^{\infty}x(t)e^{-i\omega t}dt\tag{1}$$ $$x(t)=\frac{1}{2\pi}\int_{-\infty}^{\...
user avatar
  • 79.2k
11 votes
Accepted

Hilbert transform of sinusoid -- apparent contradiction

The error lies in the assumption that if $g(t)$ is the Hilbert transform of $f(t)$, then the Hilbert transform of $f(-t)$ must be $g(-t)$. This is not the case. Let $f^-(t)=f(-t)$. Then we have $$g(...
user avatar
  • 79.2k
11 votes

Show others how I hear myself

The most practical attempt that I am aware of is by Won and Berger (2005). They simultaneously recorded vocalizations at the mouth with a microphone and on the skull with a homemade vibrometer. They ...
user avatar
  • 231
10 votes
Accepted

Feature extraction/reduction using DWT

I think it is kind'a similar to soft and hard thresholding using in wavelet de-noising. Have you come across this topic? pywt has already an in-built function for ...
user avatar
  • 10.4k
10 votes
Accepted

Show others how I hear myself

It is not impossible but it is not going to be a walk in the park too. What you would be trying to do is to add to the voice signal, those vibrations that are delivered to the ear via the bones and ...
user avatar
  • 10k
8 votes

Discrete wavelet transform; how to interpret approximation and detail coefficients?

Wavelet transforms can be more difficult to interpret than FFT at face value due to the various representations, nomenclature and output formats. I had to study more than 15 resources to get a good ...
user avatar
  • 181
8 votes
Accepted

What is a Kravchuk transform and how is it related to Fourier transforms?

Transliterations of Ukrainian names have different avatars in English (and in others languages as well). You can find Kravchuk polynomials, and other papers like On Krawtchouk Transforms or ...
user avatar
7 votes
Accepted

Whether Fourier transform formula be considered as Convolution or Correlation?

Correlation and convolution are basically the same operations. You can express the cross-correlation of two functions $f(t)$ and $g(t)$ by a convolution: $$R_{fg}(\tau)=f(\tau)\star g^*(-\tau)$$ ...
user avatar
  • 79.2k
7 votes
Accepted

$\mathcal{Z}$-transform of $\frac{1}{n^2}$

The problem is not sufficiently specified, because the range of admissible values of $n$ is missing. Here I make the assumption that we consider $n>0$. With this assumption we have $$X(z)=\sum_{n=...
user avatar
  • 79.2k
7 votes

IIR Hilbert Transformer

This is achievable with two parallel all pass filters. The two all pass filters synthesize an odd ordered low pass filter whose pass band extends from -90º to +90º in the z-domain. (I will discuss ...
user avatar
6 votes
Accepted

How condition for existence of Fourier transform is valid?

As mentioned in Batman's answer, the condition of the sequence being absolutely summable is only sufficient but not necessary. The Fourier transform can be extended to $\ell_2$ sequences, i.e. ...
user avatar
  • 79.2k
6 votes
Accepted

Hilbert transform too large to store (out of core processing)

I would use a linear phase FIR Hilbert transformer, and use block processing, such as the overlap-add method. That means that you partition the input signal into contiguous non-overlapping blocks and ...
user avatar
  • 79.2k
6 votes
Accepted

FFT equivalent for generalized unitary transforms

It's all about structure. One early paper on this is A Unified Treatment of Discrete Fast Unitary Transforms, 1977: A set of recursive rules which generate unitary transforms with a fast ...
user avatar
5 votes
Accepted

Continuous Wavelet Transform with Scipy.signal: what is parameter "widths" in cwt() function? How do time-frequency?

complex morlet was added Aug 10, 2007 ricker and cwt were added Sep 20, 2011 There's no indication that cwt is meant to be compatible with ...
user avatar
  • 14.9k
5 votes

IIR Hilbert Transformer

I have insufficient reputation to answer in the comments, so here goes: I believe Olli calculated his coefficients using some kind of genetic algorithm (I don't know the details). All I did was plot ...
user avatar
5 votes

Fourier Transform of a signal using direct integration and properties

Your first solution using the properties of the Fourier transform is correct. Your second solution is wrong, because you forgot to include the unit step function. Your function $g(t)$ should be ...
user avatar
  • 79.2k
5 votes
Accepted

Nyquist Rate (Sampling Frequency) for $ {f}^{2} \left( x, y \right) $

By the Convolution Theorem multiplication in Time / Spatial domain is equivalent of Convolution in the Frequency Domain. The sampling rate (In its classic interpretation) is proportional to the ...
user avatar
  • 39k
4 votes

What does boundary discontinuity in DFT imply?

Because the left (top) edge of an image is unlikely to be a reflection of its right (bottom) edge, there are discontinuities all along the edges of an image when it is viewed as N-periodic. These ...
user avatar
4 votes
Accepted

Transformed Direct Form II filter: Z transform: reduce multiplications

It's instructive to look at the actual difference equation implemented by the system. If you define a sequence $w[n]$ as the output of the first (left-hand side) adder you get $$w[n] = x[n] + Aw[n-1]\...
user avatar
  • 79.2k
4 votes

Connection with system analysis and laplace&Z transform

You are right that the (bilateral) Laplace transform can be interpreted as the Fourier transform of $e^{-\sigma t}f(t)$. However, I think that the significance of the Laplace transform only becomes ...
user avatar
  • 79.2k
4 votes
Accepted

Fourier Transform of exponential

The plot is of $$\mid X\left(i\omega\right) \mid = \sqrt{\left(\frac{1}{a+j\omega}\right)\left(\frac{1}{a-j\omega}\right)} = \frac{1}{\sqrt{a^2 + \omega^2}}$$ against $\omega$ In particular $\omega$...
user avatar
  • 1,017
4 votes

Finding Laplace Transform without ROC

Strictly speaking you can't because without specifying the ROC, the inverse Laplace transform is generally not unique. However, in many contexts there is the implicit assumption of causality of the ...
user avatar
  • 79.2k
4 votes

Can anyone explain how does CZT (Chirp Z Transform) really help in 'spectral zooming'?

The CZT allows for a fairly general evaluation of the Z transform - the more general evaluation path looks like a spiral, so it has a radial component step size as well an angular step size.For ...
user avatar
  • 2,636
4 votes
Accepted

Daubechies wavelet transform

Looks like you need a general explanation of the discrete wavelet transform (DWT). DWT breaks a signal down into subbands distributed evenly in a logarithmic frequency scale, each subband sampled at a ...
user avatar
4 votes
Accepted

Useful natural "Hilbert-like" $n$-uples and $n$-fold "analytic signals

The generalisation of the concept of an analytic signal is not straight forward. I'm quite certain however that looking for such a generalisation with quarternions (or even octonions) will not turn ...
user avatar
  • 4,324
4 votes

When doing a Hilbert-transformation, why not simply multiplying by an exponential?

In contrast to Jason R's answer I claim that the Hilbert transform is a phase shift by $-\pi/2$ for real-valued signals. By definition, a phase shifter shifts the phase of a sinusoidal signal by some ...
user avatar
  • 79.2k
4 votes
Accepted

How to sketch the following discrete-time signal?

I give you some hints and then you can solve this homework. Your $x[n]$ has only $8$ nonzero values. Figuring out what happens to them (in an exhaustive way) is not difficult. Consider $(n-1)^2$ and ...
user avatar
  • 4,095
4 votes
Accepted

Conditions for which the Hilbert transform returns a correct phase

A single instantaneous phase estimate may or may not make any sense if there is more than one frequency peak in the signal's local spectrum. So, to get a better single frequency and phase estimate, ...
user avatar
  • 33.8k
4 votes
Accepted

integration property of fourier series

Note that the antiderivative of a function is only defined up to a constant. Furthermore, note that if you integrate a periodic function, the result is not necessarily periodic. Let $$x(t)=\sum_{k=-\...
user avatar
  • 79.2k
4 votes

Is there an easy way to translate a Fourier transform table from angular frequency $\omega$ to Hertz $f$?

Your confusion comes from the fact that you use $X(\cdot)$ for denoting both functions, the function of $\omega$ and the function of $f$, but they are really two different functions, because $$X(\...
user avatar
  • 79.2k

Only top scored, non community-wiki answers of a minimum length are eligible