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8

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 processing, and more widely in mathematics, that are different from their everyday meanings. In signal processing "real" denotes a signal component that is in ...


5

Usually Tikhonov Regularization is applied in the following form: $$ \arg \min_{\boldsymbol{x}} \frac{1}{2} {\left\| A \boldsymbol{x} - \boldsymbol{y} \right\|}_{2}^{2} + \frac{\lambda}{2} {\left\| \boldsymbol{x} \right\|}_{2}^{2} $$ This formulation can be seen as: MAP Estimator with the prior of $ \boldsymbol{x} \sim N \left( 0, {\sigma}_{x}^{2} \right) $....


4

As mentioned in the comments you can just take the integral of the squared magnitude. Notice that the magnitude of $|e^{a+bj}| = |e^a||e^{bj}| = e^a$ for $a,b \in \mathbb{R}$, comparing with your function we see that $|x_i(t)|=1$. We conclude that $x_i(t)$ is a power signal, not an energy signal, i.e. you must define a time interval to integrate. Then the ...


3

Complex, or analytic wavelets enable: Instantaneous frequency, amplitude, and phase extraction - detailed post. Robust feature extraction for classification, stable against time-warping deformations (and, if coupled with time averaging, robust to time shifts), and averaged information recovery via higher-order transforms -- paper, lecture Exact analyticity ...


2

HINT: It is based on the fact that $$\sin(x) = \frac{e^{jx} - e^{-jx}}{2j}\quad\text{and}\quad j^2 = -1$$


2

The efficiency is in spectral utilization, at the expense of hardware complexity. Spectrum is such an expensive resource and the demand for data so great that this is a compelling trade to make. Although sampling rates can be reduced in the hardware, it takes two real data-paths (and as the OP mentioned 4 real multipliers and two adders to implement a full ...


2

The bandwidth part is somewhat trivial. If you do eg 1000 complex samples/symbols/... per second, that represents twice as much information as 1000 real samples/symbols/... per second. The convenience part depends of what you find convenient, what operations you want to do, and what capabilities the building blocks available to you have. To me, the «complex ...


2

A complex-valued analytic signal: $$\begin{align} x_\mathrm{a}(t) &= \sum\limits_{k=1}^{K} g_k \ e^{-\alpha_k t} \ e^{j(2\pi f_k t + \phi_k)} \ u(t) \\ &= \sum\limits_{k=1}^{K} \underbrace{g_k e^{j\phi_k}}_{c_k} \ e^{(-\alpha_k+j2\pi f_k)t} \ u(t) \\ &= \sum\limits_{k=1}^{K} c_k \ e^{(-\alpha_k+j2\pi f_k)t} \ u(t) \\ ...


2

It's all about vectorization. N = 8; K = 10; k = 1:K; % row vector f = k * 100; % row vector alpha = k / 10; % row vector a = k / 10; % row vector phi = k * pi; % row vector deltat = 1; n = (0:N-1)'; % column vector b = (1:N)'; % column vector x = sum(a.*exp(1j*phi).*exp((-alpha+1j*2*pi*f)*deltat.*n)+b, 2);...


2

For what values of the model parameters, does it generate a real signal? That's probably not possible. Roughly speaking, your frequencies are all positive and a real signal must have a conjugate symmetric spectrum, i.e. equal amount of positive and negative frequencies. If you make the sum run from $-K$ to $+K$ and set $a_k = a_{-k}, \alpha_k = \alpha_{-k}, ...


1

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 computers can easily handle complex numbers. The other interpretation is that a complex signal is just two real signals, one of which is labeled "imaginary&...


1

If the signal is a real audio waveform that only has AM components, the baseband equivalent waveform will be real unless the carrier has a phase offset. If the intention is to model the effect of carrier phase offset, this can be accomplished by multiplying the baseband waveform with $e^{j\phi}$ for a given carrier offset $\phi$ resulting in a complex signal....


1

There are a few things that don't add up. Firstly, if you have the sequence [0 0 0 0 1 0 0 0] mapped on your subcarriers, then you are not using all subcarriers. Second, if you are using Matlab, the 5th element of the sequence above (so 1) doesn't correspond to the first subcarrier in your OFDM signal (unless you are not doing an ifftshift before computing ...


1

just generate two independent gaussian random numbers (with a good generator like the Twister) having zero mean and identical variance, call one the real part and call the other the imaginary part.


1

It's because $$ j \cdot ( a + b) = -\frac{a + b}{j} $$ which stems from the fact that the imaginary unit $j$ has the property : $$ j = \frac{-1}{j} $$


1

The bandwidths might be the same. I agree with Jason’s assessment. Is there a difference? Yes. Let’s say we construct both signals such that their frequency domain’s are equivalent from $0-F_s$. For the real signal the frequency domain would be limited to $X(F_s-f) = X^{*}(F_s+f)$. For the complex signal it would be $X(f) = X(F_s+f)$. Can the IQ ...


1

The FFT probably had to be ran on a square image that was padded out, the inverse has returned the padded out image.


1

The simple answer is that $$e^{j\omega_0t}\neq e^{j(\omega_0+2\pi)t}\tag{1},\qquad t\notin\mathbb{Z}$$ Since $t$ is a real variable, the inequality is true for uncountably many values of $t$. Equality is only achieved for countably many integer values of $t$. From $(1)$ it follows that for real $t$, the function $e^{j\omega_0t}$ is not $2\pi$-periodic in $\...


1

HINT: You need to go back to the original definition of the Fourier coefficients via the integral, and treat $|n|=1$ as a special case. Your mistake happened there already (you divided by zero).


1

The line fscanf(sensorInput, "%lf + j*%lf\t", &x[i]); should be fscanf(sensorInput, "%lf,%lf\t", &x[i].real, &x[i].imag); Notice that C++ provides complex type, they support the common complex operations, and provide an io, and they can read complex numbers writen as (1,3) directly file_in &...


1

Here is one aspect that often creates confusion: In many cases, complex signals are used solely as a mathematical convenience, not because the complex representation is required by anything physical. A good example is circuit analysis in electronics: all physical quantities (voltages, currents, impedances) are real and you can do the analysis with these ...


1

Phasors are useful for the analysis of (real-valued) linear time-invariant (LTI) systems. A phasor is a complex number $$C=|C|e^{j\phi}=A+jB\tag{1}$$ which represents a sinusoidal signal: $$x(t)=\textrm{Re}\big\{Ce^{j\omega t}\big\}=A\cos(\omega t)-B\sin(\omega t)=|C|\cos(\omega t+\phi)\tag{2}$$ That's already the end of the story. Eq. $(2)$ is the generally ...


1

The method of subtracting the clean signal from the received signal is a good method and will work as long as the scaling is matched up. As an alternative, you can also try the cross-correlation method which takes care the of the amplitude estimate. This previous answer gives the basic idea, How to calculate time domain SNR using known sequence. If $x(t)$ is ...


1

The closed contour $C$ must lie inside the region of convergence, so for the ROC $|z|<2$ you have no poles inside $C$ for $n\ge 0$, hence $x[n]=0$ for $n\ge 0$.


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