A key objective of multi-channel FFT analysis equipment is to determine the transfer function, which represents the transfer characteristics of electrical circuits and mechanical systems.
If we denote the input and output signals to a system as x(t) and y(t), and their Fourier transforms as X(f) and Y(f), respectively, then the transfer function H(f) is defined as follows:
H (f) = Y (f) / X (f) ---------------------- (1)
Since it is a function of the frequency between the input and output, it is also called a frequency response function, but here we will explain it as synonymous with a transfer function.
As shown in definition (1), the transfer function H(f) is a complex number, as it is the complex ratio of X(f) and Y(f). If we let its real part be HR(f) and its imaginary part be HI(f), it can also be expressed in terms of amplitude (also called gain) and phase.
H (f) = HR (f) + j HI (f) ---------------- (2)
Amplitude |H (f)| = √HR (f) 2 + HI (f) 2--- (3)
Phase θ (f) = arctan(HI (f) / HR (f))------ (4)
Now, the physical meaning of a transfer function is that it represents the amplitude ratio and phase difference of input and output signals to a system at each frequency.
For example, if a signal x(t) = A sin(2π f0 t) is input to the system and a signal y(t) = B sin(2π f0 t + φ) is output, then the transfer function at frequency f0 will have an amplitude of B/A and a phase of φ (since a normal physical system is a causal system, it has a lag element and the phase is negative).
Since a transfer function is a complex function with two elements (real and imaginary, or amplitude and phase) for frequency, it cannot be represented by a single graph like real functions such as time waveforms or power spectra. There are usually three ways to represent it:
The first method, as defined in equation (2), involves using frequency as a common axis and displaying the real and imaginary parts as separate diagrams arranged vertically; this is called a Coq Sportif diagram.
The second method, as shown in definitions (3) and (4), uses frequency as a common horizontal axis and displays amplitude (magnitude) and phase as separate graphs vertically on the vertical axis; this is called a Bode plot. Since the magnitude of the amplitude graph is always a positive number, displaying it on a logarithmic scale allows for a balanced display of both large and small peaks. The frequency axis is also often displayed on a logarithmic scale.
The third method is a polar coordinate representation with the real part on the horizontal axis and the imaginary part on the vertical axis, called a Nyquist plot (vector plot). This method results in a single graph, but it has the disadvantage that the frequency axis is not clearly displayed. Therefore, the frequency axis is sometimes represented using a three-dimensional representation of the Nyquist plot. This diagram has the advantage of clearly displaying resonance points.
Transfer functions have applications in a wide range of fields, such as filter characteristics in electrical systems, frequency characteristics of speakers in acoustic systems, vibration resonance frequencies and damping characteristics in mechanical systems, and system stability in servo systems.
(Excerpt from the email newsletter issued on August 27, 2003)
