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4

This is one of the best problems to demonstrate Fourier Series properties, and specifically the time derivative property: $$\frac{d}{dt}x(t) \overset{FS}\longleftrightarrow j2\pi kf_0 X_k$$ Instead of computing the integral $$X_k=\frac{1}{T_0}\int_{T_0}x(t)e^{-j2\pi kf_0t}dt$$ which is time consuming, you can take the first derivative of your signal and ...

3

Not a full solution but a few hints: Note that your function has an odd symmetry. Hence the even Fourier coefficients should be zero. They don't seem to be in your case and I think that's a consequence of how you integrate: you integrate from 0 to $T/2$ instead of $-T/2$ to $T/2$. If you did both halves, they would turn out with opposing signs and then ...

2

To perform carrier recovery for FSK, if the data is random and can assumed to be equiprobable, then you can take the mean of the derivative of the phase versus time to determine the carrier offset. Optionally the phase change from one sample to the next can be estimated using using a complex conjugate product of successive samples (for small angles, the ...

2

To make it sound natural you typically introduce some small random pitch, speed & gain modulations. That's not a trivial amount of work, especially if you want something that sounds natural and good and maintains the original phrasing. This is a pretty common plug-in in audio processing. It's typically called "vocal doubler" or "voice doubler" and ...

2

Your problem is at this step. You're doing half of a shortcut, but not the other half: $$x_k = \frac{2A}{T} \int_{0}^{T/2} \frac{2t-T}{T} e^{-i2 \pi k f_0’ t }$$ Start with the Fourier series definition (with the notation tidied up): $$x_k = \frac{A}{T_0} \int_{-T_0/2}^{T_0/2} \frac{2t-T_0}{T_0} e^{-i2 \pi \frac{k}{T_0} t } dt$$ What you did with this ...

1

The following is what I believe to be an optimized approach for performance in the presence of additive white noise when no other information is known about quantity of pulses or their amplitude distributions, beyond that they are 100 or 200 us long rectangular pulses and repeat at the 1KHz and 1.4KHz rates. This can be even further improved if any other ...

1

"I totally understand the concept of Fourier transform" Lucky you if you really do. Some of us (me, in first place) don't (in totality). The Fourier transform (and its avatars) is a prototype for duality. Duality here means that you can represent a signal on some primal domain (time) onto a dual domain (here frequency). This transformation is meant to ...

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There are various flavors physical interpretations of the continuous Fourier Transform. Here is the one that works best for me: The amplitude of the Fourier Transform is a metric of spectral density. If we assume that the unit's of the original time signal $x(t)$ are Volts than the units of it's Fourier Transform $X(\omega)$ will be Volts/Hertz or $V/Hz$. ...

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First, the original $x$ and reconstructed $\hat{x}$ signals have a peak amplitude around $0.35$. Their difference peaks above $0.40$. That happens between two time-shifted like-alike signals. Second, the difference seems to relate to the amplitude. Processing maybe be non-linear, and it could be interesting to look at relative differences like \$2(x -\hat{x}...

1

Many people, including Andrew Ng in his Deep Learning Specialization, emphasize the importance of domain knowledge and developing hand crafted features. Only then one can achieve significant improvements in performance. A. Ng clearly talks about how hand crafted features are nowadays looked down upon but in fact, are important. Fundamental concepts in signal/...

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