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
$$g(...
- 88.9k
11
votes
Show others how I hear myself
The most practical attempt that I am aware of is by Won and Berger (2005). They simultaneously recorded vocalizations at the mouth with a microphone and on the skull with a homemade vibrometer. They ...
- 231
10
votes
Accepted
Show others how I hear myself
It is not impossible but it is not going to be a walk in the park too.
What you would be trying to do is to add to the voice signal, those vibrations that are delivered to the ear via the bones and ...
- 10.6k
9
votes
Discrete wavelet transform; how to interpret approximation and detail coefficients?
Wavelet transforms can be more difficult to interpret than FFT at face value due to the various representations, nomenclature and output formats. I had to study more than 15 resources to get a good ...
- 191
8
votes
Accepted
What is a Kravchuk transform and how is it related to Fourier transforms?
Transliterations of Ukrainian names have different avatars in English (and in others languages as well). You can find Kravchuk polynomials, and other papers like On Krawtchouk Transforms or ...
- 31.7k
7
votes
Accepted
$\mathcal{Z}$-transform of $\frac{1}{n^2}$
The problem is not sufficiently specified, because the range of admissible values of $n$ is missing. Here I make the assumption that we consider $n>0$. With this assumption we have
$$X(z)=\sum_{n=...
- 88.9k
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 ...
- 88.9k
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 ...
- 31.7k
5
votes
Accepted
Time domain maximum from frequency domain data?
Suppose that Alice has a vector $\mathrm x \in \mathbb R^n$. She computes the DFT of $\mathrm x$
$$\mathrm y := \mathrm F \mathrm x \in \mathbb C^n$$
where $\mathrm F \in \mathbb C^{n \times n}$ is a ...
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 ...
- 88.9k
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 ...
- 88.9k
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 $\omega$...
- 1,067
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 ...
- 13.5k
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 ...
- 2,841
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 ...
- 4,556
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 ...
- 88.9k
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 ...
- 4,225
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, ...
- 35.2k
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
$$x(t)=\sum_{k=-\...
- 88.9k
4
votes
Is there an easy way to translate a Fourier transform table from angular frequency $\omega$ to Hertz $f$?
Your confusion comes from the fact that you use $X(\cdot)$ for denoting both functions, the function of $\omega$ and the function of $f$, but they are really two different functions, because
$$X(\...
- 88.9k
4
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 ...
- 31.7k
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.
...
- 28k
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, ...
- 8,815
3
votes
Can anyone explain how does CZT (Chirp Z Transform) really help in 'spectral zooming'?
This part of that paper suggests the answer:
The issue being that the standard FFT just does linearly-spaced spectral samples from $-f_s/2$ to $f_s/2$ where $f_s$ is the sampling frequency. The CZT ...
- 25.2k
3
votes
Accepted
Is DCT (Discrete Cosine Transform) of Type-2 lossless or lossy?
Mostly yes, but it depends on the context. Let us elaborate.
DCT-II is one of the many forms of Discrete Cosine Transforms, and probably the most widely used one, as it is (somehow) present in JPEG ...
- 31.7k
3
votes
Fourier transform 4 times = original function (2D and higher)
The n-dimensional (discrete) Fourier transform is separable, that means the dimensions can be treated independently. So the property that the FT is a fourth root of the identity applies to n-...
- 4,556
3
votes
Accepted
Confusion in basics of Laplace Transform
The unilateral Laplace transform is used for analyzing causal linear time-invariant systems, which have an impulse response $h(t)$ that is zero for $t<0$. The unilateral Laplace transform can be ...
- 88.9k
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