Power Spectrum Estimation of three sinusoids in white noise

Question

Let's assume we have a random process consisting of three sinusoids in white noise: $$x[n] = 3 \cdot \sin(ω_1 \cdot n + ϕ_1) + 5 \cdot \cos(ω_2 \cdot n + ϕ_2) + 2 \cdot \sin(ω_3 \cdot n + ϕ_3) + v[n]$$ where $ϕ_1$, $ϕ_2$, $ϕ_3$ are uncorrelated uniformly distributed random variables in $[0,2\pi]$

I want to estimate the power spectral density of $x[n]$. For that purpose I apply two different methods:

1. Periodiagram of $x[n]$
2. Welch method (Averaged Modified Periodogram) using Hamming window

I generated the following code and ran in Matlab:

N = 1024; 
w1 = 0.3 * pi;
w2 = 0.6 * pi;
w3 = 0.7 * pi;
phi1 = unifrnd(0, 2*pi);
phi2 = unifrnd(0, 2*pi);
phi3 = unifrnd(0, 2*pi);
u = randn(1, N);
n = 0:N-1;
x = 3 * sin(w1*n + phi1) + 5*cos(w2*n + phi2) + 2*sin(w3*n + phi3) + u;

fig = figure;
plot(n,x);
title('signal x[n]');           
xlabel('samples');                          
ylabel ('Magnitude');  

fig=figure;
[p,f]=periodogram(x);
plot(f,p);

fig=figure
[pw,f]= pwelch(x, N/8, N/16);
plot(f,pw);

Which is the difference in these two methods about the resolution and the variance of the power spectrum ?

Answer 1

The standard Periodogram uses all the data and computes its power spectrum spectrum estimatation at once; therefore it provides more spectral resolution but less estimation reliability (larger estimation variance).

The Welch's modification to the standard Periodogram is the concept of dividing the signal into shorter blocks and averaging the computed per block Periodograms, and therefore it has less spectral resolution (due to shorter blocks) but provides smaller estimation variance, due to averaging.