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
$$g(...
- 91.2k
11
votes
Show others how I hear myself
The most practical attempt that I am aware of is by Won and Berger (2005). They simultaneously recorded vocalizations at the mouth with a microphone and on the skull with a homemade vibrometer. They ...
- 231
10
votes
Accepted
Show others how I hear myself
It is not impossible but it is not going to be a walk in the park too.
What you would be trying to do is to add to the voice signal, those vibrations that are delivered to the ear via the bones and ...
- 10.7k
9
votes
Discrete wavelet transform; how to interpret approximation and detail coefficients?
Wavelet transforms can be more difficult to interpret than FFT at face value due to the various representations, nomenclature and output formats. I had to study more than 15 resources to get a good ...
- 191
8
votes
Accepted
What is a Kravchuk transform and how is it related to Fourier transforms?
Transliterations of Ukrainian names have different avatars in English (and in others languages as well). You can find Kravchuk polynomials, and other papers like On Krawtchouk Transforms or ...
- 32.1k
7
votes
Accepted
$\mathcal{Z}$-transform of $\frac{1}{n^2}$
The problem is not sufficiently specified, because the range of admissible values of $n$ is missing. Here I make the assumption that we consider $n>0$. With this assumption we have
$$X(z)=\sum_{n=...
- 91.2k
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 ...
- 331
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 ...
- 111
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 ...
- 91.2k
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 ...
- 32.1k
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 ...
- 91.2k
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 ...
- 32.1k
5
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, ...
- 9,054
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 ...
- 91.2k
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 ...
- 4,305
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, ...
- 35.6k
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 ...
- 54
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
$$x(t)=\sum_{k=-\...
- 91.2k
4
votes
Is there an easy way to translate a Fourier transform table from angular frequency $\omega$ to Hertz $f$?
Your confusion comes from the fact that you use $X(\cdot)$ for denoting both functions, the function of $\omega$ and the function of $f$, but they are really two different functions, because
$$X(\...
- 91.2k
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.
...
- 28.4k
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 $I$, ...
- 32.1k
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 (...
- 91.2k
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 ...
- 4,593
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 ...
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 $\...
- 24.7k
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 ...
- 13.6k
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 ...
- 32.1k
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 ...
- 32.1k
3
votes
Hilbert transform of sinusoid -- apparent contradiction
There is not contradiction. If you are more familiar with the Fourier transform, you may remember the time reversal property: if $$\mathcal{F} [x(t)] = X(\omega)$$ then:
$$\mathcal{F} [x(-t)] = X(-\...
- 32.1k
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