Application of the Fourier Integral in Circuit Theory and Circuit Problems

The straightforward application of the Fourier integral to determine the response of a linear invariable circuit to an arbitrary impressed force is reviewed. When a Fourier integral representation of the impressed force exists and the system starts from rest, the problem is routine. Extensions to cases such as impulses, step functions, and ramp functions, for which the Fourier integral along the real axis does not converge, may be made by suitable changes in the path of integration. Boundary conditions other than zero stored energy can be included by introducing equivalent driving forces to produce the same effect as the initial voltages and currents. Noise voltages and currents in general can not be represented as Fourier integrals over all time. Representation over a finite time with a subsequent limiting process can be used to define a spectral density or mean-square amplitude in an elementary frequency interval. Spectral densities of network responses to a specified noise source are then calculable by multiplying the squared absolute value of the appropriate transfer function versus frequency by the spectral density function of the source. Knowledge of spectral densities can be used to design optimum circuits for separation of signal and noise.