# Tag Info

### Replacing "e" in Euler's formula with another number

Say you're interested in $$M^{j2\pi f_0 t}. \tag{1}$$ Note that $$M = e^{\log M},$$ so $(1)$ can be written as \begin{align} M^{j2\pi f_0 t} &= \left( e^{\log M} \right) ^ {j2\pi f_0 t} \\ &= ...
• 15.3k
Accepted

### What would be the variance for complex number?

I will focus on the reason of the factor $1/2$ and leave aside the estimation things. The exact understanding should be : if a scalar Gaussian random variable (rv) is circular symmetric, its real and ...
• 6,595

### Does it make sense to have complex numbers representing real-world audio signals?

So it seems like real-world (discrete) audio signal might have complex values when being represented digitally, No, you misunderstood that. The discrete audio time signal doesn't have non-real ...
• 31.1k
Accepted

• 24.6k

### Neural Networks and Complex Valued Inputs

The power of complex representations remains an open topic to me. I still do strive the understand Fourier transformations. An underlying question is, to me: why would complex transformations be ...
• 31.9k
Accepted

### (graphic) Relation between FFT and complex signal

Each bin in the DFT result does not represent a sinusoid but is a spinning phasor in the time domain as $x[n] = c_k e^{j\omega_k n}$. For those less familiar, the form $Ke^{j\phi}$ with real $K, \phi$ ...
• 51.9k

### Discontinuity in the angle of a complex exponential signal

That's not really a discontinuity. On a circle the two points $-\pi$ and $+\pi$ are identified: They are the same point. That is true for all $x$ and $x+n 2\pi$ for integer $n$. If you would like to ...
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### Bandwidth with complex sampling

Complex sampling does not "break" Nyquist. IQ quadrature sampling produces twice as many bits per second of information (at the same sample rate for real or complex samples), and the 90 ...
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### FFTs of a complex signal - separating the real and imaginary parts

The Fourier transform is linear, so you have that $$\mathcal{F}[a+jb]=\mathcal{F}[a]+j\mathcal{F}[b].$$ Now, $\mathcal{F}[a]$ and $\mathcal{F}[b]$ are complex, so you have that \begin{align} \text{...
• 15.3k
Accepted

### Complex IIR to Real IIR

This answer shows how to create a real-coefficient infinite-impulse-response (IIR) filter, the output of which equals the real part of the output of a given complex-coefficient IIR filter. Also an ...
• 13.5k
Accepted

### Discontinuities in the FFT

Your parameters aren't correct for producing a whole number of cycles for each component. For each $i$ the value of $\frac{\omega_{i}}{\omega_{s}} N$ has to be a multiple of $2 \pi$. Hope this ...
• 7,560

### Derivative with respect to complex conjugate

I felt I needed to write an additional answer to try to clear my mind about the question. Here is the try, step by step. Caveat: for simplicity, I used the same notation $C$ of a function of reals $u$...
• 31.9k

### For complex values, why use complex conjugate in convolution?

The use of the conjugate in the formation of the adaptive filter isn't necessary. However, if you do not write the output using a conjugate then it is quite easy to forget that the variables you are ...
• 2,871
Accepted

### Maximum likelihood estimation complexity computation

Ordinary Least Squares problem Your $$\hat x = \arg\min_x \sum_{n=1}^{N_r} \left\lvert y_n - h_n x\right\rvert^2$$ is just a way of saying $$\hat x = \arg\min_x \left\| y - Hx \right \|$$ and that is ...
• 31.1k