已知自相关函数为Acos(wt),请确定该函数的均方值X2和均方根值x

根据自相关函数公式，可以得到：X2 = ∑(x[n]-x[n-1])^2= ∑(Acos(wt[n])-Acos(wt[n-1]))^2= ∑(2A^2[1]sin^2(wt[n]-wt[n-1])) + ∑(-4A^2[1]sin(wt[n]-wt[n-1])cos(wt[n]-wt[n-1])) + ∑(4A^2[1]sin^2(2(wt[n]-wt[n-1])))= 4A^2[1] + (-8A^2[1] - 4A^2[1]) + 4A^2[1]= -8A^2[1] + 8A^2[1]= 0所以该函数的均方值X2为0。根据定义，均方根值x[n]等于信号中每个样本值的平方的平均值除以样本数再取平方根，即：x[n] = A^2[1]sin^2(wt[n]-wt[n-1])x[n]的平均值等于该信号的功率谱密度在0Hz处的值，即：x bar = (1/N)* ∑(x[n]^2/N) = (1/N)* ∑(A^2[1]sin^2(wt[n]-wt[n-1))^2/N= (1/N)* ∑(A^4_1/((w^2t[n]^2+w^2t[n-1]^2+1)^2)= (1/N)* ∑(A^4_1/((w^2t[n]^2+w^2t[n-1]^2+1)^4)= (1/N)* (A^4_1/w^2) * ((w^2t[n]^2+w^2t[n-1]^2+1) - (w^2t[n]^2+w^2t[n-1]^2+1)^3= (A^4_1/w^2) * (w^2t[n]^2+w^2t[n-1]^2+1) * (1 - (w^2t[n]^2+w^2t[n-1]^2+1)^2/3)= (64A^4_1/w^2) * (1 - (w^2t[n]^2+w^2t[n-1]^2+1)^2/3)= (64A^4_1/w^2) * (1 - ((w_1t[n] + w_2t[n-1] + 1)^2/(w^2t^2+w^2t^2+1)))= (64A^4_1/w^2) * (1 - ((w_1w_2t[n]t[n-1] + 1)^2/(w^2t^2+w^2t^2+1)))= (64A^4_1/w^2) * (1 - (w^2t[n]^2+w^2t[n-1]^2+1))= (64A^4_1/w^2)由于信号的频率高于截止频率，该函数的平均值为零。所以该函数的均方根值x[n]为：x[n] = A^2_1sin^2(wt[n]-wt[n-1])x[n]的平均值等于该信号的功率谱密度在0Hz处的值，即：x bar = (1/N)* ∑(x[n]^2/N) = (1/N)* ∑(A^2_1sin^2(wt[n]-wt[n-1))^2/N= (1/N)* ∑(A^4_1/((w^2t[n]^2+w^2t[n-1]^2+1)^2)= (1/N)* ∑(A^4_1/w^2) * ((w^2t[n]^2+w^2t[n-1]^2+1) - (w^2t[n]^2+w^2t[n-1]^2+1)^3= (64A^4_1/w^2) * (1 - (w^2t[n]^2+w^2t[n-1]^2+1)^2/3)= (64A^4_1/w^2) * (1 - ((w_1t[n] + w_2t[n-1] + 1)^2/(w^2t^2+w^2t^2+1)))= (64A^4_1/w^2) * (1 - (w^2t[n]^2+w^2t[n-1]^2+1))= (64A^4_1/w^2)