# Frequency Response and Network Analysis

You can use the following functions to characterize the frequency response of a network:

- Frequency response function
- Impulse response function
- Coherence function

## Frequency Response Function

The following figure illustrates the method for measuring the frequency response of a network.

In the previous figure, you apply a stimulus to the network under test and measure the stimulus and response signals. From the measured stimulus and response signals, you compute the frequency response function. The frequency response function gives the gain and phase versus frequency of a network. Use Equation A to compute the response function.

| (A)

where H(f) is the response function, A is the stimulus signal, B is the response signal, SAB(f) is the cross power spectrum of A and B, and SAA(f) is the power spectrum of A.

The frequency response function is a two-sided complex form, having real and imaginary parts. You can convert to the frequency response gain and the frequency response phase in the same way you compute the amplitude and phase spectrums of a power spectrum.

To convert to single-sided form, discard the second half of the response function array.

You might want to take several frequency response function readings and compute the average. Complete the following steps to compute the average frequency response function.

1. Compute the average SAB(f) by finding the sum in the complex form and dividing the sum by the number of measurements.
2. Compute the average SAA(f) by finding the sum and dividing the sum by the number of measurements.
3. Substitute the average SAB(f) and the average SAA(f) in Equation A.

## Impulse Response Function

The impulse response function of a network is the time-domain representation of the frequency response function of the network. The impulse response function is the output time-domain signal generated by applying an impulse to the network at time t = 0.

To compute the impulse response of the network, perform an inverse FFT on the two-sided complex frequency response function from Equation A. To compute the average impulse response, perform an inverse FFT on the average frequency response function.

## Coherence Function

The coherence function provides an indication of the quality of the frequency response function measurement and of how much of the response energy is correlated to the stimulus energy. If there is another signal present in the response, either from excessive noise or from another signal, the quality of the network response measurement is poor. You can use the coherence function to identify both excessive noise and which of the multiple signal sources are contributing to the response signal. Use Equation B to compute the coherence function.

| (B)

where SAB is the cross power spectrum, SAA is the power spectrum of A, and SBB is the power spectrum of B.

Equation B yields a coherence factor with a value between zero and one versus frequency. A value of zero for a given frequency line indicates no correlation between the response and the stimulus signal. A value of one for a given frequency line indicates that 100% of the response energy is due to the stimulus signal and that no interference is occurring at that frequency.

For a valid result, the coherence function requires an average of two or more readings of the stimulus and response signals. For only one reading, the coherence function registers unity at all frequencies.