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11 votes
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Is there a relation between an analytic signal (signal processing) and an analytic function (complex analysis)?

There is a relationship between these two concepts. Let the complex function $f(z)$ be analytic on and inside a simple closed curve $C$ in the complex plane. Then Cauchy's integral formula states that ...
Matt L.'s user avatar
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8 votes
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Does analytic signal have positive instantaneous frequency?

It is not the case that the instantaneous frequency of an analytic signal is always positive. In general, the instantaneous frequency can become negative, also for analytic signals. I'll show this ...
Matt L.'s user avatar
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7 votes

How to "draw" the function/wave to send symbols using QAM?

The baseband QAM signal is complex, and the only way to draw it is by doing two drawings, one for the in-phase (real) component, and one for the quadrature (imaginary) component. The passband QAM ...
MBaz's user avatar
  • 15.3k
5 votes

Does analytic signal have positive instantaneous frequency?

Okay, so let's get a little specific about the math... The Hilbert Transform: $$\begin{align} \hat{x}[n] &= \mathscr{H}\big\{ x[n] \big\} \\ \\ &= \sum\limits_{i=-\infty}^{+\infty} \frac{1-(-...
robert bristow-johnson's user avatar
4 votes
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Obtain I/Q Components from a Real Signal On the Fly - Hilbert Transform or Digital Downconversion?

[Update: I mentioned a possible +3dB processing gain by including the Hilbert Transform prior to DDC for the case of a real IF signal in the first version of this post, which @MattL questioned so I ...
Dan Boschen's user avatar
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4 votes
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I Q sampling and baseband version of analytic signal

The problem in your derivation is the way you discard the high frequency components. Note that $\hat{s}(t)$, the Hilbert transform of the bandpass signal $s(t)$, has frequency components around $\...
Matt L.'s user avatar
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4 votes
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Magnitude of the analytic signal

By definition, the magnitude of the analytic signal $s_a(t)$ can always be considered the envelope of the signal in a purely mathematical sense, simply because the real-valued signal is given by $$s(t)...
Matt L.'s user avatar
  • 90.4k
4 votes

Envelope detection in MATLAB

The Hilbert is very simple to implement with high accuracy for narrow band signals such as the OP presents (we just need to introduce a perfect quadrature phase shift and maintain the amplitude ...
Dan Boschen's user avatar
  • 52.1k
3 votes

Calculate and interpret the instantaneous frequency

Sorry to provide an answer a year after the fact, but I stumbled across this post while searching for articles on this very topic. Your questions reflect the widespread disagreements and ...
guest's user avatar
  • 31
3 votes
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Phase change rate of human speech

I will not address the question whether or not it is meaningful to compute the instantaneous frequency of a speech signal. Instead I will show you a better method for computing the instantaneous ...
Matt L.'s user avatar
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3 votes
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Separate Simple Signal into Exponential Components

Your problem seems to be finding the analytic signal to a given real signal. What you apparently need is the Hilbert transform. This will allow you to "recreate" the imaginary part of $y(t)$ from the ...
Andreas H.'s user avatar
3 votes

Definition of minimum-phase system

one thing about a non-minimum phase system (with a rational transfer function), is that it can be thought of as the series concatenation (or cascade) of a minimum-phase system, having identical ...
robert bristow-johnson's user avatar
2 votes

QAM Constellation Construction - Basic 4/8 QAM

One can choose any mapping desired from the unique data sequences to the symbol locations on the QAM constellation. It is common and usually best to Gray-code the mapping such that only one bit ...
Dan Boschen's user avatar
  • 52.1k
2 votes
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CODECS used for VoIP calls

Let me put in an unconnected form things about audio codecs. Audio can be music or speech (discluding ultrasound, sonar etc). Music is wideband and requires high-fidelity. Speech has a lower ...
Fat32's user avatar
  • 28.3k
2 votes

Discrete Hilbert transform enhancements?

Try multiplying your $h[n]$ with a Kaiser window. $$ h[n] = \frac{1}{\pi n} \big(1 - (-1)^n \big) \ w[n] $$ where the window function is $$ w[n] = \begin{cases} \frac{1}{I_0(\beta)} \, I_0\left(\...
robert bristow-johnson's user avatar
2 votes

Block diagram for a complex impulse response

A digital system with complex input and complex output has a block diagram in which the various arrows carry complex signals, each register holds a complex number, each multiplier computes the product ...
Dilip Sarwate's user avatar
2 votes
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Analytic Signal Block in Simulink Gives Reduced Amplitude for the Imaginary Part

If the filter has a fixed order then increasing the sampling rate will make its response scale with frequency, thus the transition width increases relative to the sampling frequency. For example: if $...
a concerned citizen's user avatar
2 votes

One-sided waveforms in both time and frequency?

I thought about this a bit, and I wondered whether something that numerically approximates the wished-for signal might be good. So I implemented a project onto convex sets (POCS) approach to ...
Peter K.'s user avatar
  • 25.8k
2 votes
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One-sided waveforms in both time and frequency?

The Fourier transform of a causal signal cannot be zero in any interval $[\omega_1,\omega_2]$ with $\omega_1<\omega_2$. This follows from the Paley-Wiener condition, which states that if $A(\omega)=...
Matt L.'s user avatar
  • 90.4k
2 votes
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Why use complex envelope to model raw measurements?

'Imaginary' is a misnomer; they 'exist' as much as reals do. As for the envelope: it provides a meaningful representation of phase and amplitude over time, not otherwise doable with reals alone. To ...
OverLordGoldDragon's user avatar
2 votes

Why and How the plot between the real and imaginary component of an analytic signal is in the form of an ellipse pattern?

The real and imaginary components of an analytic signal are not in an ellipse pattern. As is well established in introductory DSP books, the real and imaginary components of the analytic signal are ...
Dan Boschen's user avatar
  • 52.1k
2 votes
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Fourier transform of $|x_\mathrm{a}(t)|^2$

Your result is correct, but it can be simplified by noticing that the two integrals in your solution are actually identical, which for $f>0$ leads to $$\mathscr{F}\left\{|x_\mathrm{a}(t)|^2\right\}=...
Matt L.'s user avatar
  • 90.4k
2 votes

Alternative to Hilbert Transform

If your signal is relatively narrow band (does not occupy the entire Nyquist bandwidth) then an FIR filter would be more efficient to compute the band-limited Hilbert Transform vs an FFT which ...
Dan Boschen's user avatar
  • 52.1k
1 vote

Construct complex signal from a real-valued time series and Hilbert transform

Clearly, $A(t)$ is the analytic signal of $v(t)$. How so ? This is simply a conversion from cartesian to polar co-ordinates. Every complex signal can be written this way regardless of whether it's ...
Hilmar's user avatar
  • 45.4k
1 vote
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Envelope detection in MATLAB

Is it because I am using a Hilbert filter of order 63 ? Yes. The Hilbert Transform is difficult to implement in practice. Its impulse response is infinitely long and infinitely non-causal. That means ...
Hilmar's user avatar
  • 45.4k
1 vote
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Can a cosine modulated complex valued baseband signal be transmitted over physical channels?

There are two interpretations of this problem. One is that only real voltages and currents can exist in the real world. Complex signals exist to simplify calculations, and in the DSP realm, where ...
MBaz's user avatar
  • 15.3k
1 vote

How an algorithm for measuring the analytic signal (Hilbert transform) gives us complex value at the end?

In general for the FT, in order for the result for the DFT or IDFT to be real, the waveform MUST be real and even (symmetric about vertical axis) or imaginary and odd (anti-symmetric about the ...
Dan Boschen's user avatar
  • 52.1k
1 vote

How an algorithm for measuring the analytic signal (Hilbert transform) gives us complex value at the end?

(while we expected it to be real). We certainly did not expect this to be real. Quite on the contrary: a real time domain signal has a spectrum that is conjugate symmetric. Be zeroing out the ...
Hilmar's user avatar
  • 45.4k
1 vote

Magnitude of the analytic signal

Using your notation for the (complex valued) analytic signal; $$s(t) = x(t) + j \hat{x}(t) = |s(t)| e^{j \phi(t)} \tag{1}$$ where $x(t)$ is the message signal and $\hat{x}(t)$ is its continuous-time ...
Fat32's user avatar
  • 28.3k
1 vote

Determine the haar wavelet function is linear or nonlinear

Linearity can play at different levels in this context: The Haar wavelet functions are piece-wise affine (constant), not linear (but sometimes people call affine "linear"). Any standard ...
Laurent Duval's user avatar

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