15.7 Relationship Between Autocorrelation Function and Power Spectral Density
543
Fig. 15.8 Variation of φ(τ )
with τ
x(t) = A sin(2π f 0 t + φ)
where the amplitude A and frequency f 0 are deterministic and the phase angle φ is
random. Determine the autocorrelation function and based on this relation determine
the spectral density of the process.
The autocorrelation of the same function has been calculated in Example 15.3,
and the value is
φ(τ ) =
A
2
2
cos 2π f 0 τ
The spectral density in terms of autocorrelation is
S( f ) =
∞
−∞
φ(τ )e
−i2π f τ dτ
=
∞
−∞
A
2
2
cos 2π f 0 τ e
−i2π f r dτ
=
A
2
2
sin 2π( f − f 0 )τ
2π( f − f 0 )
15.8 Random Response of SDF Systems
It has been pointed out in Chap. 3 that there are two approaches of studying the
response of a system: they are, time domain analysis and frequency domain analysis.
These concepts are extended here for a SDF system, whose input excitation is random.