Impulse response
This time, we'll continue with the hitting test.
When a system is viewed as input → transfer system (frequency response function) → output, the transfer system can be represented by a transfer function (frequency response function).
Let X(f), Y(f), and F(f) be the Fourier transforms of the input, transfer system, and output signals, respectively.
Y(f）＝F(f)X(f) ・・・（1）
Therefore, the output signal can be obtained by multiplication. As we discussed last time, if the input frequency is known, the output can be easily calculated. The transfer function was discussed on the frequency axis, but this time let's consider it on the time axis.
In a system where an input signal x(t) and an output signal y(t) are obtained, the response of a time-invariant linear system when a unit impulse δ(t) (delta function) is applied to an input x(t-τ) that has been delayed for a time τ is called the impulse response.
Since the frequency spectrum of a unit impulse contains all frequency components equally, X(f) in equation (1) is
X(f) = 1 (= spectrum is constant at all frequencies)
Therefore
F(f)＝Y(f)/X(f)＝Y(f)
Therefore, the impulse response in the frequency domain is the transfer function itself. Even if you don't consciously think about it, you are actually measuring the impulse response when measuring the transfer function.
So, what happens in the time domain?
In particular, if we denote the impulse response as h(t), then when a unit impulse input acts at t=0, the output at that time is y(t) = h(t). The input at t=τ is x(τ), but if we consider x(t) as a collection of impulses, the input can be expressed as x(τ)dτ. The output for this input signal is
y(t)＝x(τ)dτh(t−τ)
This is the result. To summarize, it looks like this:
Input → x(τ)δ(t-τ)dτ
Output → x(τ)h(t-τ)dτ ・・・(2)
Now, since x(t) is a state where impulse signals are being input one after another, the output will be the sum of the successively occurring impulse responses.
Impulse response and transfer function when an impulse is input to a linear system.
When the input signal is considered as a pulse train, the output signal is given by the sum of the responses of the pulse train.
Therefore, the response to a continuous input signal x(t) can be obtained by integration and is given by the following equation.
y(t)＝∫x (τ) h(t-τ) dτ ・・・（3）
(∫ represents integration from -∞ to +∞; the same applies below.)
Furthermore, by performing a variable transformation from t-τ to t, it can also be expressed as follows:
y(t)＝∫x(t-τ) h (t) dτ ・・・（4）
The integrals on the right-hand side of equations (3) and (4) are called convolution integrals (or superposition integrals).
By performing a Fourier transform on both sides of equation (4) as follows, we derive the following relationship in the frequency domain.
∫y(t) exp（-j2πft）dt＝∫∫x (t-τ) h (τ) dτ exp(-j2πft) dt
Transforming t-τ to t and rearranging the right-hand side in terms of τ,
Right side = ∫∫x (t) h (τ) dτ exp {-j2πf (t+τ)} dt
＝∫∫x (t) exp (-2πft) dt h (τ) exp (-j2πfτ)dτ
＝∫X (f) h(τ) exp (-j2πfτ) dτ
Therefore
Y (f) ＝ X (f) H(f) ・・・ （5）
Since H(f) is the response when the input signal is a unit impulse, it is the same as equation (1).
We found that the convolution integral in the time domain becomes a multiplication in the frequency domain. We also learned that the transfer function is a frequency domain representation of the impulse response.
Convolution integrals involve many calculations, making it difficult to determine the output response in the time domain. However, in the frequency domain, it becomes a product of the transfer function and the power spectrum of the input signal, making it easy to determine the output response. In fact, FFT analyzers have this as an equalization function.
For example, when installing equipment on the floor, this function can be used to estimate the power spectrum of the equipment's vibrations before installation work begins. The transfer function of the equipment is measured using a vibration exciter, and the power spectrum of the floor's vibrations is also measured, and the two values are multiplied together. The calculation result is the expected vibration power spectrum of the equipment when it is placed on the floor.
Since the transfer function represents the relationship between the input and output points, this means that the vibrational power spectrum of the output point was calculated.
Furthermore, it has a function to display the impulse response using an FFT analyzer, which is obtained by performing an inverse Fourier transform on the transfer function.
When analyzing vibrations, considering the relationship between this convolution integral, the transfer function, and the time domain and frequency domain can be helpful.
(Excerpt from the email newsletter issued on August 26, 2004)
