Hot answers tagged

23 votes

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} \\ &= ...
user avatar
  • 13.9k
14 votes
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 ...
user avatar
  • 5,780
12 votes

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 ...
user avatar
12 votes
Accepted

For complex values, why use complex conjugate in convolution?

Turns out that convolution and correlation are closely related. For real signals (and finite energy signals): Convolution: $\qquad y[n] \triangleq h[n]*x[n] = \sum\limits_{m=-\infty}^{\infty} h[n-m] ...
user avatar
12 votes

Are there any real world applications for complex-valued signals or impulse responses?

Absolutely! Conjugates are mentioned in textbooks because conjugation has no effect on real signals, but it does on complex ones. This way, formulations are more general and apply to both real and ...
user avatar
  • 1,821
10 votes

Efficient Magnitude Comparison for Complex Numbers

PROLOGUE My answer to this question is in two parts since it is so long and there is a natural cleavage. This answer can be seen as the main body and the other answer as appendices. Consider it a ...
user avatar
  • 6,903
10 votes

Efficient Magnitude Comparison for Complex Numbers

You mention in a comment that your target platform is a custom IC. That makes the optimization very different from trying to optimize for an already existing CPU. On a custom IC (and to a lesser ...
user avatar
  • 231
9 votes
Accepted

What to do after this last step?

This sum appears quite often in DSP. \begin{align} \sum_{n=0}^{N-1} \exp(-j\alpha n) &\stackrel{(a)}{=} \frac{1- \exp(-j\alpha N)}{1 - \exp(-j\alpha )}\\ &= \frac{e^{-j\alpha N/2}(e^{+j\alpha ...
user avatar
  • 5,780
9 votes
Accepted

What are imaginary signals?

How can a signal be imaginary? It's not imaginary in the everyday sense of meaning something that doesn't exist. "Real" and "imaginary" have technical meanings in signal ...
user avatar
8 votes

FIR filter design for complex signal

In fact you have two signals, and it depends on what you want to achieve, but usually you would just filter both signals (the real and the imaginary part) with the same (real-valued) low pass filter. ...
user avatar
  • 80.4k
8 votes

Efficient Magnitude Comparison for Complex Numbers

1. Logarithms and exponents to avoid multiplication To completely avoid multiplication, you could use $\log$ and $\exp$ tables and calculate: $$I^2 + Q^2 = \exp\!\big(2\log(I)\big) + \exp\!\big(2\...
user avatar
8 votes

Efficient Magnitude Comparison for Complex Numbers

Given two complex numbers $z_1=a_1+jb_1$ and $z_2=a_2+jb_2$ you want to check the validity of $$a_1^2+b_1^2>a_2^2+b_2^2\tag{1}$$ This is equivalent to $$(a_1+a_2)(a_1-a_2)+(b_1+b_2)(b_1-b_2)>...
user avatar
  • 80.4k
7 votes
Accepted

Considering the FFT of Real & Complex Signals

There's nothing wrong here - complex sinusoids like your signal really have only one peak in frequency domain! This is the fundamental idea of why we use the Fourier transform for periodic (even ...
user avatar
7 votes
Accepted

Tikhonov Regularization for Complex Matrices

Usually Tikhonov Regularization is applied in the following form: $$ \arg \min_{\boldsymbol{x}} \frac{1}{2} {\left\| A \boldsymbol{x} - \boldsymbol{y} \right\|}_{2}^{2} + \frac{\lambda}{2} {\left\| \...
user avatar
  • 41.4k
6 votes

Compression algorithms specific to complex signals

Complex signals are a special case of multidimensonal signals (where the dimension is two). A lossy approach tackling compression of multidimensional signals is vector quantization. A very good ...
user avatar
  • 4,115
6 votes
Accepted

DFT of a complex sinusoid

The last expression (sum of a geometric series) is a common abuse of notations: it should have been: $N$ if $k= k_0$ $f(r)=\frac{1-r^N}{1-r}$ with $r=e^{-j 2\pi(k-k_0)/N}$ if $k\neq k_0$ Indeed, as ...
user avatar
6 votes
Accepted

How do I convert a real baseband signal to a complex baseband signal?

To convert a real signal sampled at rate $2B$ to its complex baseband representation (sampled at rate $B$), you want to map the frequency content in the range $[0, B)$ in the real signal to the range $...
user avatar
  • 23.7k
6 votes

Derivative with respect to complex conjugate

That's a trick which you will also find in a DSP context, that's why I choose to provide an answer here. It is related to the Wirtinger derivative, and you can find more details about it in this ...
user avatar
  • 80.4k
6 votes

Efficient Magnitude Comparison for Complex Numbers

I'm putting this as a separate answer because my other answer is already too long, and this is an independent topic but still very pertinent to the OP question. Please start with the other answer. A ...
user avatar
  • 6,903
5 votes

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 ...
user avatar
  • 4,324
5 votes
Accepted

FFTs of a complex signal - separating the real and imaginary parts

Below is a chart I had of "Universal Fourier Transform Properties", that apply in either direction (going from time to frequency or going from frequency to time). For example, a signal that is ...
user avatar
  • 37.7k
5 votes

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{...
user avatar
  • 13.9k
5 votes
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 ...
user avatar
5 votes
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 ...
user avatar
  • 6,903
5 votes

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 ...
user avatar
5 votes

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 ...
user avatar
  • 2,751
5 votes

Efficient Magnitude Comparison for Complex Numbers

The Sigma Delta Argument Test I came up with my own solution with the premise of resolving maximum vector magnitude (including equality) by testing the angle for quadrature between the sum and ...
user avatar
  • 37.7k
5 votes
Accepted

For Schwarz inequality for 2 signals $s_1(t)$ and $s_2(t)$, equality holds if $s_1(t) = c\cdot s_2(t)$ ($c$ is constant). Does $c$ have to be real?

Schwarz Inequality for continuous-time Complex valued functions is given as follows: $$\left|\int^{\infty}_{-\infty}f(t)^* \cdot g(t) dt \right|^2 \le \int^{\infty}_{-\infty}\left|f(t)\right|^2dt \...
user avatar
  • 2,576
5 votes

Are there any real world applications for complex-valued signals or impulse responses?

Software-defined radio (SDR) models real band-pass signals as complex baseband signals. All signals and filters operate on complex numbers.
user avatar
  • 4,721
4 votes

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 ...
user avatar
  • 34k

Only top scored, non community-wiki answers of a minimum length are eligible