PSD-Wiener Khinchin-xcorr problem due to the division coefficients of the PSD

Hi,

I am trying to calculate the PSD from the autocorrelation. I am finding the following problems doing this (it is mandatory for me using this autocorrelation):

1. The test 1 is verified. I used the same division that are placed in this documenthttp://aaronscher.com/Course_materials/Communication_Systems/documents/PSD_Autocorrelation_Noise.pdf, but unfortunately I am not sure why the psd should by divided by the length of the signal instead of the length of the autocorrelation.
2. The Parseval theorem (%parseval failed) is failed so I am wandering if this means that my division are not correct. If I apply the parseval theorem using the normalization %parseval passed, the test is passed. So can you help me to understand if my "normalization" is wrong.

I hope you can help me. I tried a lot of time to find the right normalization.

clear; clc; close all;

v_c=dlmread('v_c.txt', '\t', 1, 0)

v1=(v_c(:,2:2))

%Tinc rappresenta il passo temporale

Tinc=0.001;

Fs=1/Tinc;

r1 =xcorr(v1,v1)/Fs;

freq =transpose( -Fs/2:Fs/length(r1):Fs/2-(Fs/length(r1)));

delta_f=(freq(2)-freq(1))

PSDfromr1 = abs(fftshift(fft(r1)))*(1/((length(v1))))

%test 1 CumulatePSDfromr1 should be equal to rms2v1
CumulatedPSDfromr1=sum(PSDfromr1)*0.005

rms2v1=rms(v1)^2

% parseval failed

vf2=sum((PSDfromr1).^2);
vt=sum(r1.^2)

% parseval passed

vf=sum(abs(fftshift(fft(r1))).^2)/length(r1);


figure (1)
plot(freq,PSDfromr1)