A number of methods are available for analyzing frequency response test data. The procedures discussed in this chapter should be sufficient to supply a suitable analysis method for most situations. Methods based on analog equipment will be considered first, then methods based on digital computing equipment will be considered.

4.1. Fourier Analysis—Analog Computer Methods

An analog computer circuit for the Fourier analysis algorithm of Section 2.9 is shown in Fig. 4.1. This method is sound in principle, but problems often arise because of inaccuracies in analog multiplication.

An alternate analog computer method that eliminates the need for analog multipliers has been developed (1-3). Consider the circuit diagram shown in

4.2. The transfer functions that relate and 02 to / are

G,(s) = OJI = s/(s2 + w2) (4.1.1)

G2(s) = OJI = — co/(s2 + со2) (4.1.2)

The impulse responses corresponding to these transfer functions are

h^t) = cos cot (4.1.3)

h2(t) = —sin cot (4.1.4)

where hi{t) is the impulse response of 0, (response of Oj if an impulse were input to the circuit of Fig. 4.2), and h2(t) the impulse response of 02. These impulse responses may be used to give the responses of the system to arbitrary inputs by means of the convolution integral:

0,(t) = f cos со (t — t)/(t) dx (4.1.5)

Jo

02(t) = — f sina>(t — x)I(x)dx (4.1.6)

Jo

The following trigonometric identities may be used to cast these results into an alternate form:

sin co(t — t) = sin cot cos cox — cos cot sin cox (4.1.7)

cos co{t — x) = cos cot cos cox + sin cot sin cox (4.1.8)

The results are

Oj(t) = cos cut /(t) cos cox dx + sin cot I(x) sin cox dx Jo Jo

02(t)=—sin cut /(t) cos cox dx + cos cot I /(t) sin cox dx Jo Jo

If the signal is periodic, со = 2kn/T, and we obtain cos conT = 1, sincunT = 0

where n is the number of cycles of data analyzed, and T the period. Thus, if the integration is stopped at a time equal to an integer multiple of the period of the signal, the results are

pnT

I(t) cos cot dr

0

I(t) cos cot dt — (j/nT) I(t) sin cot dt

0 j 0 we obtain

tчіТ ’

(1/nT) I(t)e-Jo>’dt = (l/nT)Ol(nT)-(j/nT)02(nT) (4.1.13)

Jo

This method has also been implemented (2) using a digital simulation of the analog circuit of Fig. 4.2.