4

I doubt that there is such a rule because your assumption- the negative of computational work more than offsets the benefits of lower error probabilities- assumes that computations are expensive and errors are not. The "costs" of computational work and errors depends entirely on the application.


4

Samples of colored noise (taken at different times) generally are correlated random variables because the autocorrelation function of the noise process is not a delta function as it is in the case of white noise. Thus, if we assume a zero-mean process (noise is generally assumed to be regardless of its color), then the covariance of two signals separated in ...


3

The answer is no, your likelihood is not correct because $\mathbf{x}$ is a random variable. Equation 2 should be: $$ p( \mathbf{y} \mid \mathbf{x} ) \quad \text{not}\quad p(\mathbf{x}) $$ You have a prior distribution on $\mathbf{x}$. MLE is suitable for "deterministic but unknown" $\mathbf{x}$. Also, you observe $\mathbf{y}$ not $\mathbf{x}$ (or at least ...


3

Hi: I don't have time to carefully look at everything right now and I'm unfamiliar with circular complex random variables. But, if it's similar to regular normal rv's and the two variables are independent , then their sum has mean zero and their variance is $ 2 \sigma^2 + 2 \sigma^2_{v}$. So, the likelihood can be written having mean zero and that variance. ...


3

The formula you need, which I don't see in the scanned pages, is the N-dimensional Gaussian distribution of the received vector given that the signal $s_m$ was transmitted: $$f(\vec{r}|s_m)=\frac{1}{(\pi N_0)^{N/2}}\exp\left[-\frac{\sum_{j=1}^N(r_j-x_{mj})^2}{N_0}\right]\tag{1}$$ The maximum likelihood receiver seeks to maximize this conditional ...


2

The FFT is really a bank of matched filters - each FFT bin corresponds to the output of a matched filter. The criterion for a matched filter is to maximize the SNR at the output of the filter - Note SNR at the output is measured differently than at the input. While the FFT can be used for frequency estimation, it isn't great. The resolution of the FFT is $1/...


2

Coming up with a "good" estimator for a parameter of interest is not an easy task because it is important to define what good means. There are many ways of defining it, depending on your application. "Good" and "optimal" mean the same thing. Maximum likelihood (ML) estimation arises from one specific intuitively pleasing way of defining good: an estimator ...


2

Very intuitively, the Generalized Cross-Correlation is a "standard" cross-correlation of the windowed signals (I'll restrict myself to window-GCC, I'm pretty certain there's others, too!). Windowing happens to increase the "peakiness" of the cross-correlation. It's basically down to the same trade-off between temporal and spectral precision that you ...


2

AWGN is a specific noise model that is to varying physical circumstances, of varying utility. Not AWGN is not a specific noise model, but you can look at it as detection not dependent on Gaussian assumptions. There is no one kind of optimal detector for every kind of non Gaussian noise. A good book is https://books.google.com/books/about/...


2

I looked this up on Wikipedia: A complex Gaussian random variable $V = \mathfrak{Re}(V)+j\mathfrak{Im}(V)$ is said to be zero mean circularly symmetric $\mathcal{CN}(0,\Gamma)$ if the random vector $[\mathfrak{Re}(V),\mathfrak{Im}(V)]$ is a Gaussian random vector with mean $[0,0]$ and covariance matrix $ \frac{1}{2}\begin{bmatrix} \mathfrak{Re}(\Gamma) & ...


2

Hi: It's worded more clearly now in that you're estimating A and there's only one RV which makes more sense. Consider the first link I sent in the other message and go to where it says the "likelihood is". Your likelihood is quite similar except, since your variance is multiplied by 2, this causes 2 things to happen in terms of how would you change the ...


2

Normally you start with a probabilistic model of your data (observations). That model includes a noise term which has some variance or standard deviation which can vary over time or can be fixed. Then you write up the joint pdf of the data, normally written something like this $p(\boldsymbol{x}|\theta)$, where $\theta$ is a vector that includes all the ...


1

Basically each pixel is a realization so all you need is to work in the 3rd dimension (Though you can also get better by using the Spatial Data). Method 1 So the trick here is to use the multiple images to estimate the Mean (True value) of each pixel and then calculate the STD on all samples (numRows * numCols * numRealizations). Assuming we have single ...


1

Maximum Likelihood under the assumption of Additive White Gaussian Noise (AWGN) is always equivalent to finding the hypothesis with the minimum distance to given data. Since minimizing distance is equivalent (In the euclidean Space) of maximizing the correlation you can always build the idea of Match Filter for parameter estimation in the settings of ML ...


1

The probability of error of the ML detector is equal to the probability that the received vector $\mathbf{y}$ is closer to $\mathbf{x}_1$ than to $\mathbf{x}_0$, which is equal to the probability that the noise component in the direction of $\mathbf{x}_0-\mathbf{x}_1$ is greater than half of the Euclidean distance between $\mathbf{x}_0$ and $\mathbf{x}_1$. ...


1

You must attention to the written text. It doesn't say the ML isn't optimal, what it says is that the problem isn't regular LS problem but Least squares problem with Constraints. The constraints make analytic solution infeasible and hence in order to solve it one must go through any signal in the space of valid signals and mark the one which minimizes the ...


1

I'm inclined to think this is true, but so far I've only gotten a counterexample: Consider the channel: \begin{bmatrix}0.5 & 0.5 & 0 & 0\\ 0 & 0.5 & 0.5 & 0 \\ 0 & 0 & 0.5 & 0.5 \\ 0.5 & 0 & 0 & 0.5\end{bmatrix} This a typewriter channel. This matrix is not invertible (it has zero determinant, and rank 2). ...


1

1) What notations to use for probability density function is it the one below: $\mathsf{P}_y(y_n|{\mathbf{u}_n})$ $\mathsf{P}_z(z_n|{\mathbf{u}_n})$ what goes in the subscript if I want to use $z$? One way to write the density function is to subscripted upper case letters for the random variables. So $p_{Y_n}$ denotes the density of random variable $Y_n$ ...


1

There are 3 major reasons in my opinion which makes the Maximum Likelihood Estimator so popular: Intuition It is very clear what's the logic behind this method and what you maximize. It makes sense even if you have little knowledge in probability and statistics. Properties It has great properties (Some are only asymptotically, namely with many samples) such ...


1

OK, let's see if I can answer this now. The original log likelihood expression is $$ L(\mathbf{y} | \mathbf{\Theta}) = -\frac{N}{2} \ln (2\pi \sigma^2_w) - \frac{1}{2\sigma^2_w} \sum_{n=0}^{N-1} ( y_n - \mathbf{h}^T \mathbf{z}_n)^2 $$ where $y_n, n=0,1,\ldots,N-1$ are the known noisy measurements, $\mathbf{h}$ are your (unknown) filter coefficients and $\...


1

Let's have a look on the following model: $$ y \left[ n \right] = \left( h \ast x \right) \left[ n \right] + \left( g \ast w \right) \left[ n \right] $$ Where $ x \left[ n \right] $ is the signal of interest and $ w \left[ n \right] $ is the AWGN with unit Variance. In Matrix form it is written by: $$ \boldsymbol{y} = H \boldsymbol{x} + G \boldsymbol{w}...


1

I had a look at your code, and I know why you get 0 BER. The thing is that you do not add any noise, you should use Eb/N0 to calculate the noise variance, and add a complex noise on your channel output. The distortion added by "chanOutput = filter(chCoeffs,1,modSignal);" merely adds linear ISI, by convolving the channel impulse response with the input ...


1

You can think of MAP as a regularization of the ML. Just like you have regularization for Least Squares Problem (They can be built, mostly, as MAP problem). The nice thing is that, as always, the best regularization is more data, namely, in most case when there is a lot of data they collide (Namely, low sensitivity fir the Posterior PDF). So they differ ...


1

Maximum a Posteriori (MAP) is the same as Maximum Likelihood Estimation (MLE) except with a Bayesian prior distribution on whatever it is that you're trying to estimate. So if you have prior information on the distribution of point spread functions then MAP will work better.


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