What is White Gaussian Noise (WGN)?

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Iain Explains Signals, Systems, and Digital Comms

Күн бұрын

Explains White Gaussian Noise (WGN) from a Signals and Systems perspective.
** Note that I unfortunately made a minor typo when I wrote the equation for the p.d.f.: The square root symbol in the denominator of the prefactor should extend further to include the sigma squared term, or alternatively if the square root is kept as it is, then it should be just sigma, not sigma squared.
** Another point to make is that I should have mentioned that the noise power, sigma^2, that I wrote at the bottom of the page, is taken over a unit of bandwidth. In other words, it is assuming that the WGN has been sampled at the output of an ideal low pass filter, with bandwidth 1 Hz (ie. between -0.5 Hz and 0.5 Hz). This is something that is often done in textbooks without mentioning it explicitly. Of course, without considering a limited bandwidth, the noise power would be infinite, since the area under the power spectral density "flat line" is infinite. Or in other words, the height of the "delta" in the autocorrelation function is infinity (since the N_0/2 label on the delta function relates to the area of the delta, which is infinitely thin and infinitely high. For more details on the delta function, see the links below).
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Пікірлер: 53

@arsalanwasim4177 4 жыл бұрын

simple and straight forward explanation, keep up the good work, Sir!

@iain_explains 4 жыл бұрын

Glad you found it useful.

@PrashantKaswekar 4 жыл бұрын

i love pen and paper explanation! Thanks for the video!

@iain_explains 4 жыл бұрын

Glad you liked it!

@nabeelahsan504 3 жыл бұрын

Your videos are a life saver!!!

@iain_explains 3 жыл бұрын

Glad you're finding them helpful.

@ahmeda2476 3 жыл бұрын

Very useful! Thank you.

@arkopratimsen9503 3 жыл бұрын

Nice one ! Just a minor typo I noticed , to point it out : The square root in the denominator of the Gaussian distribution prefactor should include the sigma squared term or if outside the root then it should be just sigma.

@iain_explains 3 жыл бұрын

Arrgh, yes, thanks, that's annoying. I'll add a note to the description under the video. Thanks for picking it up.

@arkopratimsen9503 3 жыл бұрын

@@iain_explains it happens .. keep up the good job of making these well explained important and nice videos ....

@ananthakrishnank3208 3 ай бұрын

Neat explanation!

@iain_explains 3 ай бұрын

Glad you think so!

@amila9395 3 жыл бұрын

thank you sir. helps me a lot

@iain_explains 3 жыл бұрын

Glad to hear that

@hydersyed7998 2 жыл бұрын

As the mean is zero, the DC power(which is equal to (mean)^2) is zero. Then the total power(which is the value of the Rx(T) at T=0) is equal to the AC power(which the also the variance). Therefore, the variance of noise is equal to the value of the autocorrelation of the noise at T=0 which is No/2.

@agapaitanveermou2275 Жыл бұрын

Love you sir

@iain_explains Жыл бұрын

I'm so glad you like the videos. It's always great to hear when people find them useful/interesting.

@analysislearning9179 3 жыл бұрын

Best One

@knavaneethan79 8 ай бұрын

thank you very much

@iain_explains 7 ай бұрын

You are welcome

@user-zg6kx5sv3i 2 жыл бұрын

I like it

@nipunjindal9866 3 жыл бұрын

Sir could you please solve this question Q 1 - A signal which takes the values +A, 0 and -A volts for T seconds with equal probability, is transmitted over a channel with additive white Gaussian noise of two-sided PSD equal to η/2. An integrate-and-dump type of receiver is used. What threshold voltages should be used if the probability of the receiver committing an error is to be independent of which signal is transmitted?

@pyaysoeoo558 3 жыл бұрын

sir, please explain about gaussian

@tharinduchandraratne941 2 жыл бұрын

Sir, your videos are really helpful and thanks for uploading. I have a question: According to the autocorrelation function of white noise, variance (sigma^2) or average power is infinite for unconstrained bandwidth case. It means the standard deviation also becomes infinite. So, the Gaussian p.d.f becomes fatter (with diminishing height) as the standard deviation increases. Since the total area under the proper p.d.f. must be 1, how can we realize the p.d.f. of white noise as it's variance approaches infinity?

@iain_explains 2 жыл бұрын

Great question. It's important to note that continuous time white noise is only a mathematical construct. In practice, no physically realisable process can change _instantly_ from one value to another distinct value, potentially infinitely far away. There's a note in the description below the video that explains that the noise power that I wrote down is taken over a unit of bandwidth. In other words, it is assuming that the WGN has been sampled at the output of an ideal low pass filter, with bandwidth 1 Hz (ie. between -0.5 Hz and 0.5 Hz). This is something that is often done in textbooks without mentioning it explicitly. Of course, without considering a limited bandwidth, the noise power would be infinite (as you point out), since the area under the power spectral density "flat line" is infinite.

@tharinduchandraratne941 2 жыл бұрын

@@iain_explains Thank you for the clarification sir.

@farahnazabouk2461 3 жыл бұрын

سلام!متشکرم .عالی بود.⚘

@iain_explains 3 жыл бұрын

Thanks so much. I'm glad you found the video helpful.

@analysislearning9179 3 жыл бұрын

I have a question please, the power of the White noise is given by its variance as its becomes equal to PSD. If the mean is not zero, will we get an additional term in PSD as DC component?

@iain_explains 3 жыл бұрын

Good question. I've got an upcoming video that explains more about noise power, so please keep a look out for that in the next week or so. But in answer to your specific question: If the random variables in the random process have a non-zero mean, then yes, there would be a delta function in the PSD at f=0.

@kiriakospipigkas5090 Жыл бұрын

Hello sir, may i ask you, why we corrupt the signal with zero-mean white noise with a variance of 4 in the FFT method?

@iain_explains Жыл бұрын

I don't see how your question relates to my video, sorry.

@shoroukraafat8929 3 жыл бұрын

what is the relation between the independence of noise at different time values and the conversion to a Dirac-delta function? I didn't get that part at 2:20 . . is N0/2*delta is a known auto-correlation function for a certain case or something?

@iain_explains 3 жыл бұрын

If a random process is 1. stationary, 2. has independent samples from one sample time to the next, and 3. has samples from a distribution with zero mean, then the autocorrelation function is a delta function. This is the case for noise. R_N(tau) = E[N(t)N(t+tau)] which is the definition of the autocorrelation function = E[N(t)] E[N(t+tau)] since N(t) is independent of N(t+tau), and since E[N(t)]=0 since it is zero mean, therefore R_N(tau) = 0 for all values of tau, except for tau=0. For tau=0, R_N(0) = E[N(t)^2], which is the variance of N(t).

@shoroukraafat8929 3 жыл бұрын

@@iain_explains Thanks a lot. It's much clearer now.

@usmanzafar4751 2 ай бұрын

@@iain_explainsMy question may seem silly. But why Variance of Noise process is No/2…?

@mustaphaalkhafaaf5512 3 жыл бұрын

could you please explain what is the second moment and why is it called the second moment , thanks.

@iain_explains 3 жыл бұрын

E[X^n] is the n-th moment of the random variable X. If the pdf of X is zero mean, then the "second moment" and the "variance" are the same.

@thanhnguyenchi2356 Жыл бұрын

I have some questions: 1. What does that the autocorrelation function result equal N0/2 at tau = 0 mean? 2. Why Sn(f) = F.T. [Rn(tau)] ?

@iain_explains Жыл бұрын

Hopefully these videos will help: "What is Power Spectral Density (PSD)?" kzfaq.info/get/bejne/etWDf7B2qtWViWQ.html and "Autocorrelation and Power Spectral Density (PSD) Examples in Digital Communications" kzfaq.info/get/bejne/jr2pp7Z9v8C0YXU.html

@eswnl1 9 ай бұрын

You define the autocorrelation as RN(tau) = E[N(t) N(t + tau)]. Another definition of autocorrelation I have seen is Rxx(tau) = 1/T Integral (X(t) X(t+ tau)). I am wondering what the difference is?

@iain_explains 9 ай бұрын

In the definition you gave, the integral is over the time variable, t. If the random process is ergodic, that definition will give the same answer as the definition I gave. If the process is not ergodic, then you'll need to use my definition. These videos will hopefully help: "What does Ergodic mean for Random Processes?" and "Are Stationary Random Processes Always Ergodic?" kzfaq.info/get/bejne/pdSordhj3bu1ZHk.html

@mayurshankar3181 3 жыл бұрын

Sir, why dont you start a telegram channel or something... We would be glad to get daily updates about the topics!! #sogood!

@malifsyahputranasution7650 Жыл бұрын

sir.. what is the relationship between SNR (signal-to-noise ratio) and variance? matlab have function awgn (for additive, white, gaussian noise) with input SNR and can return variance as the result. Instead of asking us to give variance as the input for function awgn, matlab then asking us to give the SNR as the input, and based on the SNR, the appropriate variance can be obtained. for example, SNR 30 dB gives us variance equal 0.001, SNR 20 dB gives variance equal 0.01, for now I just understand, if SNR high, variance will be low, and vice versa.. I mean the correlation in mathematical representation like your explanation in this video, thank you sir, this question just in case you can explain because your video make me clear about the 'white' in awgn function matlab, but just one more question in my head

@malifsyahputranasution7650 Жыл бұрын

btw nice video sir.. thank you so much about explanation the 'white'

@iain_explains Жыл бұрын

Hopefully this video helps: "What is Noise Power in Communication Systems?" kzfaq.info/get/bejne/ldeeZ7Wrsti8cpc.html

@malifsyahputranasution7650 Жыл бұрын

@@iain_explains thank you sir

@TranMinh-dc5kn 7 ай бұрын

hi sir, can you explain to me what correlation function is ? Thank you

@iain_explains 6 ай бұрын

The correlation function is best explained via its special case of Autocorrelation: "What is Autocorrelation?" kzfaq.info/get/bejne/nrWmeJuh2qi-e2g.html and this video might also help "How are Correlation and Convolution Related in Digital Communications?" kzfaq.info/get/bejne/jctlpJh3r8nGk4k.html