# ARMA models for non stationary signals

let us suppose that we have non stationary signal,whose value is given by

56.69
69.22
-2.19
21.80
9.85
-80.51
-56.28
-18.79
-14.32
12.26
74.71
67.37
24.43
-9.05
-47.30
-62.60
-47.64
-26.36
20.78
67.58
49.83
28.24
25.14
-43.10
-72.81
-17.82
-12.38
-34.82
58.96
69.83
-5.10
14.65
28.29
-64.48
-61.23
26.43
1.90
-47.52
51.01
41.97
-41.54
-0.62
60.86
-52.44
-37.61
61.99
-10.36
-60.20
30.45
22.95
-55.33
34.50
85.71
-13.38
-12.63
54.70
-37.86
-77.38
4.65
9.28
-24.83
68.53
85.37
-15.80
-7.33
4.31
-72.06
-73.52
1.21
17.90
16.23
63.14
63.68
-14.40
-22.79
-46.29
-73.40
-35.51
13.92
27.94
65.99
50.76
6.08
-5.44
-33.80
-73.27
-14.35
14.32
-10.47
41.59
65.77
-9.65
-17.11
25.02
-34.94
-57.50
32.16
35.22
-47.44
14.10
68.68
-35.63
-23.43
52.82
-18.17
-45.41
35.20
13.02
-77.34
9.09
54.62
-41.86
5.54
100.61
-2.50
-47.15
21.43
-30.39
-105.88
10.12
54.47
-15.25
51.19
85.27
-13.40
-63.08
-3.18
-55.73
-85.04
18.21
56.47
24.51
54.11
55.81
-22.11
-48.48
-52.59
-52.83
-32.17
21.34
52.16
57.12
32.52
3.33
-26.44
-45.33
-51.45
-26.23
22.70
7.25
30.30
49.79
10.05
-47.82
7.81
5.60
-52.47
12.12
49.13
-27.13
-11.40
53.12
-14.12
-37.25
64.87
24.10
-64.07
16.83
25.67
-84.88
-23.01
76.72
2.82
-18.01
81.46
19.46
-69.46
-4.69
-15.00
-95.40
-14.27
81.06
10.99
12.24
77.50
-13.87
-80.90
-39.37
-33.24
-55.32
27.60
87.91
46.31
35.38
34.22
-36.01
-70.78
-43.57
-18.78
-5.34
48.43
53.05
39.49
20.91
-11.03
-39.82
-42.87
-34.33
-21.77
33.55
22.34
11.90
27.41
7.81
-40.11
1.82
9.00
-45.68
-10.45
44.91
-7.91
-47.04
41.03
24.43
-45.47
28.51
54.19
-51.30
-33.24
27.62
-63.64
-53.94
84.27
31.39
-19.69
67.12
39.14
-85.28
-46.64
3.16
-58.10
-30.32
96.07
56.72
2.62
46.12
4.95
-114.65
-43.85
0.83
-27.53
16.15
99.72
40.78
-9.38
15.97
-32.67
-78.52
-36.29
8.67
3.76
36.16
61.85
31.10
-9.09
-14.47
-51.55
-35.03
-16.66
0.81
39.57
42.17
-8.22
13.39
20.67
-36.92
-21.53
25.32
-13.28
-27.59
41.31
-3.05
-56.63
21.87
41.42
-24.58
19.23
65.43
-55.32
-57.21
28.64
-12.84
-50.87
75.03


i have tried to apply autoregressive model to this data,by using following matlab code

[Pxx,f]=pyulear(B,20,1024,100);


plot(f,Pxx)

and i have got following picture

in this case i had not any information about order,just i have took randomly,now one point is that this signal is represented by following form

$y[t]=A_1(sin(\omega_1*t+\phi_1)+A_2*sin(\omega_2*t+\phi_2)+....+A_p*sin(\omega_p*t+\phi_p)$+$z(t)$

where $\phi$ are some constants and also frequencies and amplitudes,$z(t)$ is white noise,because this signal is nonstationary,can we apply standard autoregressive models(AR,ARMA) to this model?which methods are suitable for better spectral resolution?i am not interested estimation of parameters,just i am concerned about spectral resolution,i want to distinguish from each other closed spaced frequencies,or when $\omega_p\approx \omega_{p+1}$thanks in advance

• Why do you think the signal is non-stationary? – Peter K. Mar 28 '14 at 13:25
• amplitude,phases and frequencies are constants,are not they non stationary? – dato datuashvili Mar 28 '14 at 13:31
• If $A_k$, $\omega_k$ and $\phi_k$ are all constants (for $k=1\ldots p$) then the signal is stationary. If $A_k = A_k(t)$ (i.e. is time-varying) then the signal may be non-stationary. – Peter K. Mar 28 '14 at 13:34
• but i have hear that for example in such model if phases are uniformly distributed,then signal is stationary right?but there they are constants – dato datuashvili Mar 28 '14 at 13:36
• Both can be true: the phases can be uniformly distributed, but will generally be constant for a single realization of the signal. The idea is that we don't know what the $A_k$, $\omega_k$ and $\phi_k$ are, so we have to model that statistically (with a distribution). Once we start measuring a particular signal, though, all of those values are now constants. – Peter K. Mar 28 '14 at 13:38

$$y[t]=A_1(\sin(\omega_1 t+\phi_1)+A_2 \sin(\omega_2 t+\phi_2)+....+A_p \sin(\omega_p t+\phi_p) + z(t)$$
Your statement i am not interested estimation of parameters,just i am concerned about spectral resolution seems like an oxymoron. I think what you mean is that you're not concerned with estimating $A_k$ and $\phi_k$, but you are interested in $\omega_k$ for the case where $\omega_k \approx \omega_{k+1}$.