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Questions tagged [complex]

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8
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3answers
2k views

Replacing “e” in Euler's formula with another number

Does Euler's formula remain valid if we use any real number other than the constant $e$? For example replacing $e$ with 5 would make the formula look like this: $5^{it}$. I tried this idea in ...
8
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2answers
328 views

For complex values, why use complex conjugate in convolution?

Taken from Adaptive Filter Theory (2014) written by Haykin page 110 : $$y(n) = \sum_{k=0}^{\infty} w_k^*u(n-k), \quad n=0,1,2,...$$ where $u$ and $w$ are complex values. My question is why use ...
6
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2answers
3k views

Even and odd signal energy property

In Signals and Systems by A. V. Oppenheim, A. S. Willsky, S. Hamid Nawab, 2nd Edition, and Signals and Systems, Simon Haykins, Barry Van Veen, 2nd Edition there is a problem related to energy of real-...
5
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2answers
789 views

DFT of a complex sinusoid

I'm attending this course (Coursera: Audio Signal Processing for Music Applications) in which the professor derives a general equation for Discrete Fourier Transform (DFT) for a complex sinusoid. The ...
5
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1answer
999 views

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

I have radio telescope observations that have resulted in two real-valued signals (corresponding to the right- and left-handed circular polarizations). The signals are sampled at rate $2B$, and ...
5
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3answers
3k views

FIR filter design for complex signal

I have a complex signal generated by an impedance analyzer. What is the best approach for designing a low pass FIR filter for this? Is a real filter applied separately to the real and imaginary ...
4
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5answers
2k views

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

In this course (Coursera: Audio Signal Processing for Music Applications) the professor uses an example of obtaining the DFT of a complex sinusoid: \begin{align} x_2[n]&=e^{\jmath\left(2\pi f_0 n+...
4
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2answers
833 views

Discontinuities in the FFT

So I am taking the Fast Fourier Transform of the following function: $$ x[n] = \displaystyle\sum\limits_{i=0}^{5} A_{i} \cos\left(\frac{\omega_{i}}{\omega_{s}} n + \phi_{i}\right) $$ Where the ...
4
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2answers
2k views

Neural Networks and Complex Valued Inputs

[not sure if this or stats.stackexchange was the correct location for this post, so put it on both for now.] I've seen some recent papers describing complex valued neural networks like this one: Deep ...
4
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1answer
258 views

FIR design for arbitrary magnitude and phase: Why can I separate real and imaginary parts like this?

In this question about the design of a FIR filter with arbitrary magnitude and phase specifications user robert bristow-johnson suggested to split the desired complex frequency response $H$ into its ...
3
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1answer
3k views

What would be the variance for complex number?

When $x$ is a zero mean random variable then $$\sum_{n=1}^N x_n x_n^T = N \sigma^2_x\,\text,$$ where the variance is $\sigma^2_x$. I'm considering Complex Normal Distributions where the real and ...
3
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3answers
573 views

Discontinuity in the angle of a complex exponential signal

Plotting this complex expontential: $$z[n]=100e^{j0.1n},\quad\text{we have a discontinuity at}\quad \arg[n]=10\pi$$ I guess this is related to the angle formula: $$\theta=\arctan\left(\frac{\Im\{z[...
3
votes
3answers
1k views

Derivative with respect to complex conjugate

I have a real function $C$ of a complex vector $x$. While taking the gradient of the function $C$ for minimising the same, why do we take the derivatives with respect to the complex conjugate of $x$, ...
3
votes
1answer
619 views

Complex IIR to Real IIR

I have created an IIR design algorithm that generates complex coefficients (there is no symmetry in the poles and zeros). However, the IIRs will be used to filter a real signal. Is there a closed form ...
3
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4answers
3k views

On the meaning of s-plane and it's link to a transfer function

Considering Fourier analysis and let's say I'm walking on the blue frequency axis in the below 3D plot from zero towards infinity: So each time I encounter a non zero blue bar, I check the frequency ...
3
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2answers
2k views

Finding the fundamental frequency of a periodic signal

Suppose we have the signal $$x(t) = e^{j\omega_1 t} + e^{j\omega_2 t} + e^{j\omega_3 t},$$ where all the frequencies are rationally related (that is, the ratio of any pair of frequencies is a rational ...
2
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2answers
118 views

What to do after this last step?

I am solving a question from book in which I have to use summation. It is as follows: $$ \frac{1}{10}\sum_{n=0}^{9} e^{-jk\omega_0n} $$ The value of $\omega_0$ is $\frac{2\pi}{10}$. What I ...
2
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2answers
4k views

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

I have a complex time varying signal at a single frequency x = a + jb where a represents the contribution from the cosine basis ...
2
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3answers
681 views

Compression algorithms specific to complex signals

I am looking for (lossy or lossless) compression algorithms dedicated to complex signals. The latter could be composite data (like the left and right for stereo audio), a Fourier transformation, an ...
2
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2answers
7k views

Considering the FFT of Real & Complex Signals

I've been implementing a website to perform the FFT of various signals, real & complex. Examining the first example, a real signal $x[n] = 10 cos(2\pi\times4n)$, I got the following FFT: Which ...
2
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1answer
89 views

complex multiplier in divide and combine FFT

I am studying radix 2 algorith from Proakis' book. But I'm a bit confusied why 1st DFT $G_1$ is not multiplied by complex entity while 2nd DFT $G_2$ is being multiplied by complex entity $W$ as shown ...
2
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1answer
127 views

Does a linear phase FIR filter shifted in frequency preserve linear phase?

Let's say I have a symmetric (and therefore linear phase) FIR low pass filter with real coefficients. If I then shift this filter in some direction in frequency by multiplying its coefficients with a ...
2
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2answers
85 views

How to test I/Q modulation with only one branch (I or Q)?

Given is a direct-conversion I/Q up- and downconverter system. Receiver and transmitter share the same (10MHz) reference and hence the LO frequency is identical and there is an (unknown) phase ...
2
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2answers
4k views

Bandwidth with complex sampling

On the transmit side, I have a 20 MHz carrier frequency carrying a signal with bandwidth of 40 MHz (so 0 Hz to 40 MHz, center at 20 MHz). On the receive side, I have a dual channel ADC with each ...
2
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1answer
516 views

Spectrum of Cosine in Complex Form

The complex exponential form of cosine $$\cos(k \omega t) = \tfrac{1}{2} e^{i k \omega t} + \tfrac{1}{2} e^{-i k \omega t}$$ The trigonometric spectrum of $\cos(k \omega t)$ is single amplitude of ...
2
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3answers
2k views

Where and how do an Image and a complex number meet?

I am trying to implement a Gabor Filter bank. Gabor Filter uses a complex function. How do the filters like Gabor Filter operate on an Image? Does a complex number represent an entire Image or only ...
2
votes
1answer
108 views

Destructive interference on the Autocorrelation of the time signal of periodic CPFSK-signals.

An FSK-signal with a frequency-shift $\Delta F$, a symbol-stream $x$, $x(n) \in {\{-1,1}\} $ and symbol duration $T$ has the complex envelope: $$f[x](t)=\exp\left(2 \pi i \Delta F T \left(\sum_{l=0}^{...
2
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3answers
1k views

How do you save/convert an image to be in the frequency domain?

Quick Note: I don't believe this is a subjective question... however if it is I will gladly modify to be more objective In ImageJ, if I take the FFT of an image, how can I save this new frequency ...
2
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1answer
70 views

Difference equation with complex zero

Let's assume I have the following transfer function: $$ H(z)=\frac{z-\left(\frac{1}{\sqrt{2}}+i \cdot \frac{1}{\sqrt{2}}\right)}{z} $$ It looks like a first order highpass-filter with a complex zero ...
1
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5answers
2k views

Equation for impulse train as sum of complex exponentials

Could someone please break down what's going on in this equation for me? I understand what the left side looks like, but not so much how the right side is the same thing. Impulse train: $$\sum_{m=-\...
1
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2answers
64 views

Fractional powers of complex numbers (DSPrelated computation)

I am puzzled by this computation: https://www.dsprelated.com/showarticle/754.php (c.f. quote) Raising $ i $ to integer powers results in traversing the unitcircle in the same number of quarter ...
1
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3answers
133 views

How the FFT takes a cosine or sine and outputs the frequencies of the complex form?

If i take the Fast Fourier Transform (FFT) of a cosine function, what has turned this cosine function into its complex exponential form which consists of $e^{i \omega t} + e^{-i \omega t}$ ? Because ...
1
vote
3answers
485 views

How should this fft output be intepreted?

I am currently reading some code, so I can understand how this FFT output is actually been stored. I have a function which computes FFT of audio file, and stores is in the format. ...
1
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1answer
78 views

Maximum likelihood estimation complexity computation

I have a basic question about maximum likelihood (ML) estimator and its implementation. I am trying to simulate a communication system, while using ML at the receiver side to find the transmit ...
1
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1answer
97 views

Poles of the transfer function in Z Transform

Given that: $H(z)$ has 4 poles maximum. $H(z)$ has a pole at $z_1=a+bi$ Given that the impulse response $h[n]$ is: Symmetric: $h[n] = h[-n]$ Real: $\forall$$n$ , $h[n]$$\in$$\mathbb{R}$ How we can ...
1
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1answer
64 views

Doubts and some confusion on variance for complex rv

This question is in continuation of the one asked here. Let's say that the measurement noise $w$ or any random variable is circularly Gaussian complex. If the imaginary and real components each has ...
1
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1answer
1k views

Autocorrelation and Power Spectral Density (Discrete)

The Autocorrelation, $\phi_{aa}[\kappa]$, of a discrete time random process, $a[k]$, is defined as: $$ \phi_{aa}[\kappa] = \mathrm{E}\left\{ a[k+\kappa]a^*[k] \right\} $$ Taking its fourier ...
1
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1answer
246 views

How do you show the Discrete Time Fourier Transform of $x[n]=\cos(2\pi f_0n)$ is $ \frac{1}{2} \delta(f+f_0) + \frac{1}{2}\delta(f-f_0)$? [duplicate]

How do you show the Discrete Time Fourier Transform of $x[n]=\cos(2\pi f_0n)$ is $ \frac{1}{2} \delta(f+f_0) + \frac{1}{2}\delta(f-f_0)$? Here is my thought process: The definition of DTFT in my ...
1
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2answers
98 views

Convolution of real with complex signal

Let $x[n]$ be real signal and $y[n]=\exp(j3\pi n)$ be a complex signal Would the convolution between those two signals be $$x[n] * \Re(y[n]) + jx[n]*\Im(y[n])$$?
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1answer
65 views

Question regarding DTFT of a complex signal

I have been doing DTFT practice problems for my DSP course, and I encountered this problem in the textbook that completely stumped me. The question asks to find the DTFT of the shown signal and to ...
1
vote
2answers
151 views

MMSE - How to minimize a complex error with respect to a set of real parameters

Suppose there's a complex signal $X(k)$ (where $k \in \{0, 1, 2,...,N - 1\}$) corrupted by additive complex noise. Its estimate $\hat{X}(k)$ is a linear combination of a set of real parameters $A_r$ ($...
1
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1answer
216 views

How to extract single side of signal spectrum in SIMULINK

Signal spectrum have two side, positive and negative. I want to make these separate in two signal, by MATALB SIMULINK. But how? I can't find it's block on DSP or communication toolbox. I found a block ...
1
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1answer
86 views

Is the calculation of variance correct from an estimator — confusion regarding complex number where I am going wrong?

I have an expression for the varaince of measurement noise obtained from an estimator. The measurement noise is additive white gaussian having values in complex domain. I have squared the term $(.)^2$,...
1
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1answer
43 views

Why does the FFT of a complex variable create negative frequencies?

So I have a demodulator that streams out X and Y values. I use a spectrum analyzer within this demodulator instrument which plots the |FFT(X+iY)| against the frequency domain, which shows up with ...
1
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2answers
51 views

How would one design a (quasi) linear phase adaptive notch filter for a single complex tone?

While IIR notch filters are attractive, I need to retain phase linearity at the filter output. I imagine that it's possible to use a standard IIR notch filter: https://www.researchgate.net/...
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0answers
35 views

Threshold for CAF Surface

I am calculating a time partitioned Cross Ambiguity Function (CAF) by adding the surfaces of different time-sectioned CAFs together. Meaning, I calculate a CAF using 10 seconds of IQ data, calculate a ...
1
vote
1answer
94 views

Is there a technical term for the sum I + Q

Most likely a very simple question, but I haven't been able to find a good answer. When working with analytical signals in general, and software defined radio in particular, a common operation is to ...
1
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1answer
3k views

Matlab: What is the proper way to calculate mean square error for complex numbers

I am confused regarding the calculation of mean square error involving complex numbers. Considering, the true channel coefficients to be ...
1
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1answer
558 views

Smoothing complex data by convolution

I need to smooth noisy complex data with a Gaussian filter. Right now, I apply the filter to real and imaginary part of the data separately, which needs two convolutions. The intended results are the ...
0
votes
2answers
63 views

MLE formulation — confusion regarding the terms in the equation (Part1)

If $v \sim CN(0,2\sigma^2_v)$ is a circularly complex Gaussian random variable which acts as the measurement noise in this model $$y_n = x_n + v_n \tag{1} $$ where $x \sim CN(0,2\sigma^2)$, then is ...