frequency response

The gain and phase response of a circuit or other unit under test at all frequencies of interest. Although the formal definition of frequency response includes both the gain and phase, in common usage, the frequency response often only implies the magnitude (gain).

The frequency response is defined as the inverse Fourier Transform of the Impulse Response of a system.

Measurement

Frequency response measurements require the excitation of the unit under test(UUT) with energy at all relevant frequencies. The fastest way to perform the measurement is to use a broadband excitation signal that excites all frequencies simultaneous, and use FFT-based techniques to measure at all of these frequencies at the same time. Noise and non-linearity is best minimized by using random noise excitation, but short impulses or rapid sweeps (chirps) may also be used.

When the desired resolution bandwidth of interest is less than about 100 kHz, the fastest way to measure the frequency response functions is to use an FFT-based technique.

The book Random Data, Analysis and Measurement Procedures by Bendat and Pierson, is considered a definitive work on the error estimation techniques for the various classes of measurements (Second Edition/Third Edition) . The mathematical convetions and symbols used by Bendat and Piersol are used in this glossary.

For proper measurement, it is also important to understand the nature of the type of signals that you are dealing with. Please see Signal Types.

Units

• dB V vs. Hz
• rad (or degrees) vs. Hz
  

Measurement Approaches

Sine Generator/Voltmeter

You apply a sine wave to the input of the system under test, and measure the output voltage. Then, repeat this process for each frequency. The gain of the system is the ratio of the output voltage to the input voltage. 

Sine Generator/Phase Meter (for Phase Measurements)

Apply a sine wave to the input, and measure the phase of the output relative to the input at each frequency of interest.

This method has the advantage of being low-cost and simple. It is also quite slow, and the following assumptions must be fulfilled in order for the measurement to be accurate:

1. The output voltage of the signal generator is stable during the entire measurement, and also at all frequencies. If there is doubt about this, the voltage must be measured at each frequency.
2. The system does not create significant distortion.
3. There is no significant noise on the output of the system. Otherwise, the measured output voltage will be too high.
   As a rule of thumb, if there is 1% distortion or noise in the system, the error will be of the same order of magnitude.
4. The output must be statistically correlated to the input. This assumption is normally true in high fidelity analog systems. However, in mechanical systems, as well as systems with complex transmission mechanism (RF) and/or with digital encoding, echo cancelling, and other adaptive techniques, this assumption may not be fulfilled.

To account for all of the above, you can use digital signal processing techniques, including FFT and cross spectral methods.

Swept Sine Excitation

Use a swept sine wave generator and an associated voltmeter. You must set the averaging time of the voltmeter so that it is less than the dwell time in a given frequency range.

This method requires for you to meet the following criteria:

1. The sweep time for a given bandwidth must be greater than the reciprocal of the desired bandwidth. For example, if a 100-Hz resolution is desired, the sweep time for the 100 Hz must be at least 10 ms.
2. The integration time of the voltmeter must be short enough to the 10 ms dwell time, otherwise it cannot respond fully.

This is a variant of the first method in that it uses continuous swept sine waves, instead of discretely stepped sine waves. This variation can be faster than the Sine Generator / Phase Meter method described above, but it must fulfill the same assumptions.

Certain instruments may have "adaptive sweep," where the sweep rate adapts to the rate of change of the output signal. For example, when sweeping through a very sharp resonance, the sweep rate is reduced to fully resolve the resonance peak.

Click on demo for an interactive demonstration using the LabVIEW Player.

Swept Sine with Tracking Filtering

Similar to the Swept Sine Excitation method described above, but this method has the advantage of being able to reject noise and distortion from the system by using a filter on the output that follows the frequency of the input.

This method must meet all the requirements regarding dwell time and averaging times.

A sophisticated variant of this method offsets (delays) the receiving filter with a fixed frequency offset (corresponding to a fixed time delay), and makes it possible to measure the frequency response of delayed signal paths. For example, in acoustic applications, you can measure the frequency response of the signal reflected from the ceiling.

Sensors

When making this measurement, you should make sure that the output impedance of the sine generator is low compared to the input impedance of the unit under test. Otherwise, the actual input voltage applied may drop, or be changed as a function of frequency.

Transient or Noise Excitation with Cross Spectral Techniques

You can use any signal that contains frequency components in the range of interest. The signals aren't required to have the same amplitude. However, all measurements using Cross Spectral Techniques require simultaneous measurement of both input and output signals, using simultaneously sampling A/D Converters.

The frequency response can be computed as

where is the cross spectrum and is the autospectrum of the input.

This technique computes the correlation between the input and output signal (as a function of frequency) and hence, rejects noise and distortion. The more statistical samples that are included in the averaging, the greater the noise and distortion rejection and hence, the greater the accuracy of the measurement. The resulting statistical function, called the cross spectrum, is then normalized for the actual amplitude of the signal at each frequency on the input (called the autospectrum, or more commonly, the averaged spectrum). This gives the Frequency Response Function (FRF), which contains both magnitude and phase information. The magnitude is typically shown on a logarithmic Y axis (in dB), and the phase is often shown on a 0 to 360 degree scale. In systems with output noise, the most accurate evaluation of resonance peaks is made using the H2 method of frequency response computation, whereas the H1 technique gives the best response for anti-resonances. H2 is also useful when inadequate resolution is used in the measurement of a resonancel.

However, since the phase often shifts thousands of degrees, a technique called phase unwrapping is used, to remove the discontinuities every time the phase jumps from 360 to 0 degrees. A demo of frequency response with phase measurement can be seen by clicking demo for use with the LabVIEW Player.

This approach has the advantage of overcoming noise, distortion, and non-correlated effects. It also corrects for any loading effects on the input to the system. In addition, the technique can be extremely rapid, because it measures all frequencies of interest simultaneously. Its only weakness is that its signal-to-noise ratio can be lower than the swept sine with tracking filter technique.

Click on demo for an interactive demonstration of frequency response (gain only, no phase) using the LabVIEW Player.

Naturally Occurring Excitation

Sometimes you cannot insert an excitation signal into the system to be tested. However, if you want to measure the Frequency Response Function of a shock absorber in a car, you can use the naturally occurring "input signals" coming from bumps in the road as the excitation signals. Using cross spectral techniques, you can measure the input signal on the axle of the wheel and cross correlate it with the output signal picked up on the automotive chassis. Since the bumps are transients, they have relatively broad frequency components and make a broadband measurement possible.

When making this measurement, you should take extreme care to account for triggering and windowing conditions, and also consider potential time delays between the input and output. Thus, this technique is only recommended for experienced professionals with a thorough understanding of digital signal processing techniques.

Additional References

• Advanced Frequency Response Demonstration
  
  

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