Using the Transfer Function to Determine the Output
The transfer function H(f ) of a filter is shown in Figure 6.4. [Notice that the magnitude |H(f )| and phase ∠H(f ) are shown separately in the figure.] If the input signal is given by

v_in(t) = 2 cos(2000πt + 40° )

find an expression (as a function of time) for the output of the filter.

Question

Using the Transfer Function to Determine the Output
The transfer function H(f ) of a filter is shown in Figure 6.4. [Notice that the magnitude |H(f )| and phase ∠H(f ) are shown separately in the figure.] If the input signal is given by

v_in(t) = 2 cos(2000πt + 40° )

find an expression (as a function of time) for the output of the filter.

Accepted Answer

By inspection, the frequency of the input signal is f = 1000 Hz. Referring to Figure 6.4, we see that the magnitude and phase of the transfer function are |H(1000)| = 3 and ∠H(1000) = 30° , respectively. Thus, we have

[latex]H(1000)=3\angle 30^\circ =\frac{\mathrm{V_{out}}}{\mathrm{V_{in}}}[/latex]

The phasor for the input signal is V_in = 2∠40°, and we get

[latex] \mathrm{V_{out}} =H(1000) imes \mathrm{V_{in}}=3\angle 30^\circ imes 2\angle 40^\circ=6\angle 70^\circ [/latex]

Thus, the output signal is

[latex] \mathrm{v_{out}}(t)=6\cos{(2000\pi t+70^\circ)} [/latex]

In this case, the amplitude of the input is tripled by the filter. Furthermore, the signal is phase shifted by 30°. Of course, this is evident from the values shown in the plots of the transfer function at f = 1000.

Question 6.1: Using the Transfer Function to Determine the Output The tran...

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