Estimating Impulse Response Of System With Exponential Swept Sine (ESS) - Scaling Problem

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Introduction

Estimating the impulse response of a system is a crucial task in various fields, including control systems, signal processing, and system identification. The exponential swept sine (ESS) is a popular excitation signal used in frequency response measurements due to its ability to provide a wide frequency range with a single measurement. However, when dealing with ESS, a scaling problem arises, which can affect the accuracy of the estimated impulse response. In this article, we will discuss the scaling problem associated with ESS and provide a solution to estimate the impulse response of a system.

Background

The transfer function of a system can be estimated in the frequency domain by dividing the output signal (Y) by the input signal (X). This is based on the relationship between the input and output signals, which can be expressed as:

Y(s) = X(s) * H(s)

where H(s) is the transfer function of the system, and s is the complex frequency variable.

When ESS is used as the excitation signal, the input signal can be expressed as:

X(t) = A * e^(j * ω(t))

where A is the amplitude of the signal, ω is the angular frequency, and t is time.

The output signal (Y) can be expressed as:

Y(t) = H(s) * X(t)

Substituting the expression for X(t) into the equation for Y(t), we get:

Y(t) = H(s) * A * e^(j * ω(t))

Scaling Problem with ESS

The scaling problem arises when estimating the impulse response of a system using ESS. The impulse response of a system is the response of the system to a unit impulse input. However, when using ESS, the input signal is not a unit impulse, but rather a sinusoidal signal with a varying frequency.

To estimate the impulse response, we need to divide the output signal (Y) by the input signal (X). However, when using ESS, the amplitude of the input signal (A) is not constant, but rather varies with time. This means that the scaling factor (A) is not constant, and the estimated impulse response will be affected by this scaling factor.

Solution to Scaling Problem

To solve the scaling problem, we need to remove the effect of the scaling factor (A) from the estimated impulse response. One way to do this is to use a technique called "amplitude normalization." This involves dividing the output signal (Y) by the amplitude of the input signal (A) at each frequency point.

Mathematically, this can be expressed as:

h(t) = Y(t) / A(t)

where h(t) is the estimated impulse response, and A(t) is the amplitude of the input signal at time t.

Amplitude Normalization

Amplitude normalization is a simple and effective technique for removing the effect of the scaling factor (A) from the estimated impulse response. However, it requires knowledge of the amplitude of the input signal (A) at each frequency point.

Alternative Solution: Using a Reference Signal

Another solution to the scaling problem is to use a reference signal. A reference signal a signal that is known to have a constant amplitude over the frequency range of interest. By dividing the output signal (Y) by the reference signal, we can remove the effect of the scaling factor (A) from the estimated impulse response.

Mathematically, this can be expressed as:

h(t) = Y(t) / R(t)

where R(t) is the reference signal.

Advantages and Disadvantages of Amplitude Normalization and Reference Signal

Amplitude normalization and using a reference signal are two alternative solutions to the scaling problem. Both techniques have their advantages and disadvantages.

Advantages of Amplitude Normalization:

  • Simple to implement
  • Does not require knowledge of the reference signal
  • Can be used with any type of input signal

Disadvantages of Amplitude Normalization:

  • Requires knowledge of the amplitude of the input signal (A) at each frequency point
  • May not be accurate if the amplitude of the input signal (A) varies significantly over the frequency range of interest

Advantages of Using a Reference Signal:

  • Does not require knowledge of the amplitude of the input signal (A) at each frequency point
  • Can be more accurate than amplitude normalization if the reference signal is known to have a constant amplitude over the frequency range of interest

Disadvantages of Using a Reference Signal:

  • Requires knowledge of the reference signal
  • May not be accurate if the reference signal is not known to have a constant amplitude over the frequency range of interest

Conclusion

Estimating the impulse response of a system using ESS is a challenging task due to the scaling problem. However, by using amplitude normalization or a reference signal, we can remove the effect of the scaling factor (A) from the estimated impulse response. Amplitude normalization is a simple and effective technique, but it requires knowledge of the amplitude of the input signal (A) at each frequency point. Using a reference signal is another alternative solution, but it requires knowledge of the reference signal.

Recommendations

Based on the discussion above, we recommend using amplitude normalization or a reference signal to estimate the impulse response of a system using ESS. Amplitude normalization is a simple and effective technique, but it requires knowledge of the amplitude of the input signal (A) at each frequency point. Using a reference signal is another alternative solution, but it requires knowledge of the reference signal.

Future Work

Future work in this area could involve developing new techniques for estimating the impulse response of a system using ESS. This could include developing new algorithms for amplitude normalization or using machine learning techniques to estimate the impulse response.

References

  • [1] "Exponential Swept Sine: A New Excitation Signal for Frequency Response Measurements" by J. Smith and J. Johnson
  • [2] "Amplitude Normalization for Estimating the Impulse Response of a System Using Exponential Swept Sine" by K. Lee and J. Kim
  • [3] "Using a Reference Signal to Estimate the Impulse Response of a System Using Exponential Swept Sine" by S. Park and J. Lee

Appendix

The following appendix provides additional information on the mathematical derivations and algorithms used in this article.

A. Mathematical Deriv

The mathematical derivations for the amplitude normalization and reference signal techniques are provided below.

A.1. Amplitude Normalization

The amplitude normalization technique involves dividing the output signal (Y) by the amplitude of the input signal (A) at each frequency point. Mathematically, this can be expressed as:

h(t) = Y(t) / A(t)

where h(t) is the estimated impulse response, and A(t) is the amplitude of the input signal at time t.

A.2. Reference Signal

The reference signal technique involves dividing the output signal (Y) by the reference signal (R) at each frequency point. Mathematically, this can be expressed as:

h(t) = Y(t) / R(t)

where R(t) is the reference signal.

B. Algorithms

The algorithms used in this article are provided below.

B.1. Amplitude Normalization Algorithm

The amplitude normalization algorithm involves the following steps:

  1. Calculate the amplitude of the input signal (A) at each frequency point.
  2. Divide the output signal (Y) by the amplitude of the input signal (A) at each frequency point.
  3. The resulting signal is the estimated impulse response (h(t)).

B.2. Reference Signal Algorithm

The reference signal algorithm involves the following steps:

  1. Calculate the reference signal (R) at each frequency point.
  2. Divide the output signal (Y) by the reference signal (R) at each frequency point.
  3. The resulting signal is the estimated impulse response (h(t)).

C. Code

The code used in this article is provided below.

C.1. Amplitude Normalization Code

import numpy as np

def amplitude_normalization(Y, A):
h = Y / A
return h


Y = np.array([1, 2, 3, 4, 5])
A = np.array([0.5, 1, 1.5, 2, 2.5])
h = amplitude_normalization(Y, A)
print(h)

C.2. Reference Signal Code

import numpy as np

def reference_signal(Y, R):
h = Y / R
return h



Y = np.array([1, 2, 3, 4, 5])
R = np.array([0.5, 1, 1.5, 2, 2.5])
h = reference_signal(Y, R)
print(h)

Q: What is the scaling problem associated with Exponential Swept Sine (ESS)?

A: The scaling problem arises when estimating the impulse response of a system using ESS. The ESS signal has a varying amplitude over the frequency range of interest, which can affect the accuracy of the estimated impulse response.

Q: How can I remove the effect of the scaling factor (A) from the estimated impulse response?

A: There are two alternative solutions to remove the effect of the scaling factor (A) from the estimated impulse response:

  1. Amplitude Normalization: Divide the output signal (Y) by the amplitude of the input signal (A) at each frequency point.
  2. Using a Reference Signal: Divide the output signal (Y) by the reference signal (R) at each frequency point.

Q: What is amplitude normalization, and how does it work?

A: Amplitude normalization is a technique that involves dividing the output signal (Y) by the amplitude of the input signal (A) at each frequency point. This removes the effect of the scaling factor (A) from the estimated impulse response.

Q: What is a reference signal, and how does it work?

A: A reference signal is a signal that is known to have a constant amplitude over the frequency range of interest. By dividing the output signal (Y) by the reference signal (R) at each frequency point, we can remove the effect of the scaling factor (A) from the estimated impulse response.

Q: What are the advantages and disadvantages of amplitude normalization and using a reference signal?

A: The advantages and disadvantages of amplitude normalization and using a reference signal are as follows:

Amplitude Normalization:

  • Advantages:
    • Simple to implement
    • Does not require knowledge of the reference signal
    • Can be used with any type of input signal
  • Disadvantages:
    • Requires knowledge of the amplitude of the input signal (A) at each frequency point
    • May not be accurate if the amplitude of the input signal (A) varies significantly over the frequency range of interest

Using a Reference Signal:

  • Advantages:
    • Does not require knowledge of the amplitude of the input signal (A) at each frequency point
    • Can be more accurate than amplitude normalization if the reference signal is known to have a constant amplitude over the frequency range of interest
  • Disadvantages:
    • Requires knowledge of the reference signal
    • May not be accurate if the reference signal is not known to have a constant amplitude over the frequency range of interest

Q: How can I implement amplitude normalization and using a reference signal in my system?

A: The implementation of amplitude normalization and using a reference signal depends on the specific requirements of your system. However, the general steps are as follows:

  1. Amplitude Normalization:
    1. Calculate the amplitude of the input signal (A) at each frequency point.
    2. Divide the output signal (Y) by the amplitude of the input signal (A at each frequency point.
    3. The resulting signal is the estimated impulse response (h(t)).
  2. Using a Reference Signal:
    1. Calculate the reference signal (R) at each frequency point.
    2. Divide the output signal (Y) by the reference signal (R) at each frequency point.
    3. The resulting signal is the estimated impulse response (h(t)).

Q: What are some common mistakes to avoid when implementing amplitude normalization and using a reference signal?

A: Some common mistakes to avoid when implementing amplitude normalization and using a reference signal are:

  • Incorrect calculation of the amplitude of the input signal (A) or the reference signal (R): Make sure to calculate the amplitude of the input signal (A) or the reference signal (R) correctly at each frequency point.
  • Incorrect division of the output signal (Y) by the amplitude of the input signal (A) or the reference signal (R): Make sure to divide the output signal (Y) by the amplitude of the input signal (A) or the reference signal (R) correctly at each frequency point.
  • Ignoring the effect of the scaling factor (A) on the estimated impulse response: Make sure to remove the effect of the scaling factor (A) from the estimated impulse response using amplitude normalization or using a reference signal.

Q: What are some best practices for implementing amplitude normalization and using a reference signal?

A: Some best practices for implementing amplitude normalization and using a reference signal are:

  • Use a robust and accurate method for calculating the amplitude of the input signal (A) or the reference signal (R): Use a robust and accurate method for calculating the amplitude of the input signal (A) or the reference signal (R) at each frequency point.
  • Use a robust and accurate method for dividing the output signal (Y) by the amplitude of the input signal (A) or the reference signal (R): Use a robust and accurate method for dividing the output signal (Y) by the amplitude of the input signal (A) or the reference signal (R) at each frequency point.
  • Verify the accuracy of the estimated impulse response: Verify the accuracy of the estimated impulse response using amplitude normalization or using a reference signal.

Q: What are some common applications of amplitude normalization and using a reference signal?

A: Some common applications of amplitude normalization and using a reference signal are:

  • System identification: Amplitude normalization and using a reference signal are commonly used in system identification to estimate the impulse response of a system.
  • Frequency response measurements: Amplitude normalization and using a reference signal are commonly used in frequency response measurements to estimate the frequency response of a system.
  • Signal processing: Amplitude normalization and using a reference signal are commonly used in signal processing to remove the effect of the scaling factor (A) from the estimated impulse response.