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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(...
Matt L.'s user avatar
  • 90.5k
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 ...
StrongBad's user avatar
  • 231
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 ...
A_A's user avatar
  • 10.7k
9 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 ...
khuang834's user avatar
  • 191
8 votes
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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 ...
Laurent Duval's user avatar
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=...
Matt L.'s user avatar
  • 90.5k
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 ...
Robby Wasabi's user avatar
6 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 ...
Ross Wilkinson's user avatar
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 ...
Matt L.'s user avatar
  • 90.5k
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 ...
Laurent Duval's 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 ...
Matt L.'s user avatar
  • 90.5k
5 votes
Accepted

Question about Hilbert transform

TLDR: if the time variable $t$, and its dual variable ($f$, $\tau$) in the expression of the bivariate kernel have the same homogeneity, (I believe that) you can call it a time-domain ...
Laurent Duval's user avatar
4 votes
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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 ...
Jazzmaniac's user avatar
  • 4,583
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 ...
Matt L.'s user avatar
  • 90.5k
4 votes
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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 ...
msm's user avatar
  • 4,295
4 votes
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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, ...
hotpaw2's user avatar
  • 35.4k
4 votes
Accepted

What is the difference between Constant-Q Transform and Wavelet Transform and which is better

A Constant Q transform is a variation on the DFT. In other words, it is a type of wavelet transform. I only have a casual understanding of both types of transforms myself, so take what I'm saying with ...
Lowell Camp's user avatar
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=-\...
Matt L.'s user avatar
  • 90.5k
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(\...
Matt L.'s user avatar
  • 90.5k
4 votes
Accepted

1D DCT matlab code

You have mistyped the formula, replace this line sum = sum + y(i).*(cos((pi.*(2.*y(i)+1).*u(j))/(2*N))); with the one below, and it works fine. ...
Fat32's user avatar
  • 28.3k
4 votes
Accepted

How to alleviate the edging effect of the Hilbert transform?

The effect can be alleviated with appropriate padding, which imposes a 'statistical prior' (i.e. assumption). No padding is equivalent to periodic padding$^{1}$, meaning signal's right joins its left, ...
OverLordGoldDragon's user avatar
3 votes
Accepted

involutory transformations - why are they not so much used in signal processing?

You don't chose transforms by whether they are involutions or not. If invertibility is of interest, any simple form of inverse is sufficient. Useful transforms reveal structure of some sort or ...
Jazzmaniac's user avatar
  • 4,583
3 votes
Accepted

DFT-like transform using triangle waves instead of sin waves

The answer to this question is yes. There exist a fast triangle transform, FTT, for triangle waves which has a complexity of $N\log_2(N)$, where $N$ is the number of elements. It works the same like ...
Hans Petter Selasky's user avatar
3 votes

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

A Hilbert transform is not a phase shift of $-\frac{\pi}{2}$. As you noted in the question, its frequency response shifts positive frequencies by $-\frac{\pi}{2}$ and the negative frequencies by $\...
Jason R's user avatar
  • 24.7k
3 votes

Time domain maximum from frequency domain data?

It's generally not possible to compute the exact maximum value, but you can compute a bound on the maximum value. Assuming your data are discrete-time, and you're using the discrete Fourier transform (...
Matt L.'s user avatar
  • 90.5k
3 votes
Accepted

How to find out if a transform matrix is separable?

I admit I did not really thought about it before. I hope my notations won't be too sloppy. I assume that given an operator matrix $A(u,v)$, you can apply this operator as a transform on an image $I$, ...
Laurent Duval's user avatar
3 votes

IIR Hilbert Transformer

I did use Differential Evolution to calculate the coefficients. But you can re-design the filter pair easily using the HIIR library by Laurent de Soras (its source code will automatically unzip to a ...
Olli Niemitalo's user avatar
3 votes
Accepted

Implementing Continuous Wavelet Transform

In 1D, some of the standard references are: Continuous wavelet transform with arbitrary scales and $O({N})$ complexity, A. Muñoz and R. Ertl\'e and M. Unser, Signal Processing, 2002 A fast ...
Laurent Duval's user avatar
3 votes
Accepted

What is the meaning of negative second for a Morlet wavelet?

Continuous wavelets with symmetric envelope are often described, by convention, on a symmetric time interval: $[-T,T]$. The Gaussian being of infinite support, this means it is truncated. This is ...
Laurent Duval's user avatar

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