# Why is noise a power signal?

1. I'm generating a power signal:

Fs = 1000;            % Sampling frequency
T = 1/Fs;             % Sampling period
N = 1000;           % Length of signal
t = (0:N-1)*T;        % Time vector

X = sin(2*pi*(10)*t);


The aplitude spectrum of this signal:

Y = 1/N * fft(S); % power signal, scaling 1/N
Y1 =  abs(Y);
Y2 = Y1(1:N/2+1);
Y2(2:end-1) = 2*Y2(2:end-1);
plot(f,Y2)


If I increase the signal length to 10000, amplitude stays the same (at 1)

2. Now an energy signal:

X = ((1./(t+1).^20)).*sin(2*pi*(10)*t);


The aplitude spectrum of this signal:

Y = fft(S); % energy signal, scaling 1
...


Again: if I increase the signal length to 10000, amplitude stays the same (around 50)

3. Now a random signal:

X = randn(size(t));


If I increase the signal length to 10000, amplitude spectrum stays the same only with

Y = 1/sqrt(N) * fft(S); % scaling 1/sqrt(N)
...


It seems that noise signal is something between power and energy signal, but in literature it is called a power signal.

• "Noise" isn't defined enough. Noise can mean a random power signal, or a random energy signal, or neither, depending on how you defined your noise. You are, however, considering a finite discrete time signal and do a DFT on it: your signal is either a constant 0 or a power signal by definition. (i.e. NO, you are not generating an Energy signal, sorry.) – Marcus Müller Jan 25 at 13:31
• oh, and there's nothing "between power and energy signal". These terms are mathematically strictly defined, and you can either fit the definition, or not. – Marcus Müller Jan 25 at 13:32
• 1.I now that DFT is assuming a periodical signal, but if I increase N to 1E4,1E5, 1E6, 1E7, 1E8, the frequency amplitudes stay constant > 0. Thus, I think, for N -> Infinity it will also stay at the same constant>0. This means the spectrum is showing energy, not power. 2. If FFT scaled with 1 is shows energy, FFT scaled with 1/N shows power, I'm curious what 1/sqrt(N) scaling means. – Tycho Jan 25 at 14:06
• no, that is mathematically not correct. The DFT will always give you a PSD, not an ESD estimate. You're absolutely right, you assume the signal repeats periodically, so after your 1e8th sample which is nearly 0, you get large amplitudes again. The energy in the periodic repeated signal would be infinite, and thus, the thing can't be an energy signal. You are right, if we think of what you want to show using digital signal processing as the "real-world, continuous-time" process that you mean, then that is an energy signal. It's just that your assumption that you can represent the signal as – Marcus Müller Jan 25 at 14:07
• discrete samples, either in time or frequency domain, only works if you make your signal periodic. So, can't be an energy signal, because a periodic signal would have an infinite amount of non-zero values, and thus infinite energy. – Marcus Müller Jan 25 at 14:11