# Tag Info

8

Is it possible to reconstruct the original pure signal? No, that is information-theoretical impossible. Also, that signal doesn't exist, probably, to begin with ;) However, you can definitely increase the the SNR simply by averaging; that becomes pretty obvious when you consider the signal of interest to be correlated within your recording, whereas your ...

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Common Approaches for Commercial Denoisers Commercial denoisers are different than what you'd see on most papers. While on papers the results are mostly using objective metrics (PSNR / SSIM) and are evaluated vs. Additive White Gaussian Noise (AWGN) with high level of noise real world images are mostly with moderate level of noise with Mixed Poisson ...

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There can't be. One man's signal is another man's noise. In fact, a communication system making the absolute most of a bandwidth would be spectrally white, just like white noise, and hence be indistinguishable from noise to anyone but the receiver for that specific system.

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If you are simply interested in plotting the data then any data reduction technique would do,even if it appears to be crude. Effectively, the plotting function itself will not plot all the data, because the space assigned to the plot has a finite number of $N_x \times N_y$ pixels assigned to it. For example, if your plotting area was $1024 \times 768$, then ...

4

This could happen as discriminator gain is increased with a filter discriminator approach since in many of those approaches the gain would be maximum and linear for small signals only and then the slope of the discriminator slowly goes down coinciding with the results in your plot (such that you no longer get a perfect sine wave out for a sine wave in—- so ...

3

A continuous-time white noise process $\{X(t)\colon -\infty < t < \infty\}$ is a hypothetical construct that we can treat (in the simplified versions that we use on dsp.SE) as a zero-mean wide-sense stationary process with autocorrelation function $K\delta(\tau)$ where $\delta(\cdot)$ is the Dirac delta. More strongly, all the random variables $X(t)$ ...

3

A) The quantization of the photocurrent is an actual physical phenomenon. I'll use $\Delta T$ in place of your $dT$. It is not a variable describing the actual physical process but rather the time step of your approximate model. Effectively you are filtering the continuous-time Poisson point process by a continuous-time "moving average" filter with a ...

3

This is an excellent series of questions, so I will have a go at part of it! I will start with an example from Verdeyen’s book (J. T. Verdeyen, Laser Electronics, Prentice-Hall, Inc., Englewood Cliffs, NJ, ©1981, Chapter 14). Assume $\lambda$ = 500 nm, quantum efficiency = 0.15, photomultiplier (PMT) gain = 1.68x$10^7$, transimpedance = 1 k$\Omega$ and RC= ...

3

To be honest, I don't think CNNs, RNNs and LSTM are useful for this kind of problem – a bandpass filter followed by a threshold would be. Now, that would have three parameters: Lower cutoff frequency Upper cutoff frequency threshold value and what is usually called "Machine Learning" is nothing but finding local minima over some (loss) function with real ...

3

As Stanley Pawlukiewicz said: even under ideal circumstance, you can gain 3 dB of SNR per doubling of recordings. I.e., to increase SNR by, say, 15 dB, you'd need to average $$2^{\frac{15}{3}} = 2^{5} = 32$$ recordings. That alone shows that the whole thing isn't really practical: it just doesn't do much unless you use a crazy-high number of recordings. “...

3

Does mixing brown noise and white noise create pink noise? No. Pink noise has a spectrum of that falls with 3dB/octave (or 10dB/decade). The spectrum of the sum of white and brown noise will be "brown" at low frequencies and "white" at high frequencies. The spectrum will have two slopes: below the transition frequency it will be -6dB/octave and above it, ...

2

This is a really nice problem. Problem Formulation I will formulate it as following: Let $x \in \mathbb{R}^{n}$ be a signal. Given $y \in \mathbb{R}^{n}$ which is a noisy measurement of $x$ such that $y = x + v$ and $z$ be a noisy measurement of the derivative of $x$ such that $z = F x + w$ where $F$ is the finite differences operator. ...

2

Any filter whose frequency response magnitude is above unity at a given frequency $\omega$, amplifies signals at that frequency. In discrete-time, the typical white noise is a wideband signal whose power spectrum ranges from $-\pi$ to $\pi$. The digital first difference filter is an LTI system with the impulse response $h[n] = \delta[n] - \delta[n-1]$ and ...

2

How did the professor state this? Did they say that the noise is iid? That means that the samples are independent, identically distributed. That means they are independent by definition (they are assumed to be so). White noise, while a little more vague, also makes the assumption of independence. Again, this means that the noise is assumed to be ...

2

When would it be beneficial to model the signal as an outcome of a stochastic process? When the process that generates the image/signal has a strong element of chance. This is not related to noise necessarily. Sometimes, a deterministic model for the behaviour of a quantity simply does not exist. For example, modern supermarkets have barcode readers that ...

2

yes, you are right. But there is misconception that figure of merit is the ratio of output to input SNR. It is actually the ratio of output SNR of a receiver to the output SNR of a baseband system (without modulation). Here, the output SNR of a baseband system is taken as a benchmark for judging noise performance of receiver. Hence, FoM = (...

2

Without knowing the context, here is what probably is meant: 3.75 refers to a more real-world sensor. The sensor described in 3.74 apparently is assumed to be perfect, ie whenever there is a negative/positive sensor data, the true signal does have the associated property. Ideal sensors, however, do not exist and 3.75 tells you, that your sensor does have ...

2

Working with your definitions: $$v \left( \left( n + 1 \right) {T}_{s} \right) - v \left( n {T}_{s} \right) = \int_{0}^{ \left( n + 1 \right) {T}_{s} } g(u) du - \int_{0}^{ n {T}_{s} } g(u) du = \int_{ n {T}_{s} }^{ \left( n + 1 \right) {T}_{s} } g(u) du$$ So basically we have integration (Which is a Low Pass Filter) of White Noise over a Time Interval ...

2

The frequency domain conjugate multiplication (correlation) of the received signal with the reference signal followed by the power delay profile will provide you the overall signal to noise ratio. As explained in paper titled "SNR Estimation based on Sounding Reference Signal in LTE Uplink". There are various noise reduction algorithm exist. Which will ...

2

The key issue here is whether the signal and noise are uncorrelated. Assuming two real random variables x and y are both zero mean, the power of the combined signal is $E\{(x+y)^2\} = E\{x^2+2xy+y^2\} = E\{x^2\}+E\{y^2\}+2E\{xy\}$ If the two variables are uncorrelated, then $E\{(x+y)^2\}=E\{x^2\}+E\{y^2\}$ In other words, the power of the combined signal ...

2

Synthetic Method: If $\{\hat X(t)\}$ and $\{\hat Y(t)\}$ are zero-mean uncorrelated low-pass WSS processes with identical autocorrelation function $R(\tau)$ and identical power spectral density $S(f)$ enjoying the property that $S(f) = 0$ for $|f|>B$, then $$\hat{N}(t) = \hat X(t)\cos(2\pi f_ct) - \hat Y(t)\sin(2\pi f)ct$$ is a band-pass process whose ...

2

...And now for a differing opinion.... The OP's representation of bandpass white noise as $$n(t) = n_I \cos(2\pi f_ct) - n_Q \sin(2\pi f_ct)\tag{1}$$ is inadequate; because each sample path of this noise process is a pure sinusoid of fixed frequency $f_c$ Hz which is not noise-like at all. Why so? Well, a sample path is what one gets when all the random ...

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There are two mistakes in your code/method. The first is the term $\sqrt{\Delta t}$ in your second formula; it should be replaced by $\Delta t$. The second is in the computation of the power spectrum from the estimated auto-correlation. What you do is square the result of the FFT Y to obtain mY, but that's not correct. First of all, Y is complex-valued, and ...

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There are indeed many peak detection algorithms, and no clear consensus on which ones are "good" or "bad". But for what it's worth, your approach makes sense. Using median or other quantiles to detect sparse signals is common, e.g. the "median clipping" stage in Lasseck (2014), Large-scale identification of birds in audio recordings. In effect, you're ...

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I'd do some small adjustments to your idea (You really nailed them). Assumptions The Signal Model - Signal + Additive White Gaussian Noise (AWGN) Probably we could generalize it more but this is beyond the scope of this question. The DFT of the signal contains Peaks with relatively small roll off This is important as we're almost saying the Signal is a ...

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Assuming that the sensors share the same characteristics, have the same timing (acceleration signals are aligned), the model with $y_1= x + n_1$ and $y_2= x + n_2$, $n_1$ and $n_2$ being uncorrelated noises of the same power, averaging them is a way to reduce the noise. The theory that asymptotically, averaging $N$ sensors reduce the variance by a factor of $... 2 How to calculate the variance of the noise samples$n[j]$in terms of$N_0$and$B$, where$n[j]$=$n_f(jT_s)$and$T_s$is the sampling period? Do you know how to calculate the variance of the process$\{n_f(t) \colon -\infty < t < \infty\}$? No? Hint: it is the area under the power spectral density curve of$\{n_f(t)\colon -\infty < t < \...

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To offset or not is simply in your definitions on how you want to do the math involved and can be convenient for further processing. Subtracting a constant, or otherwise scaling the waveform, does not change the signal to noise ratio. Correlation is to multiply and accumulate, and the cross-correlation and auto-correlation functions show this correlation ...

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