Signal and noise response of a spectrum expanse

A spectrum expander samples an analog input signal periodically, stores these samples cyclically in a series of digital registers, and reads out these samples cyclically at a higher rate. An analysis of the signal and noise response of this device is given here. The above spectrum expander is precisely equivalent to an analog system consisting of 1) a filter-bank spectrometer, whose filter transfer functions are sine functions of frequency
(\sin c x\equiv \sin \pi x/ \pi x)
spaced at the Nyquist interval; 2) a modulator bank that separates the filter bank outputs, spacing them equally in frequency; and 3) An adder, that sums the modulator bank outputs. Such a spectrum expander may be followed by an appropriate physical, fixed channel-bank spectrometer, and used to determine approximately the power spectrum of noise inputs with a variety of bandwidths. The input bandwidth can be accommodated to the fixed channel-bank spectrometer by adjusting the expansion ratio. Stating this another way, such a system is equivalent to a channel-bank spectrometer with variable resolution. Moreover, such a spectrum expander may be used for real-time signal processing, since it offers a digital means of dissecting a signal in the frequency domain. Each of the above applications suffers from impairments. The present analytical results permit us to evaluate the errors in various cases of interest. These include the effects of a finite number of digits in the storage registers (equivalent to quantizing the analog input) for Gaussian noise inputs. By using a weight function on the output samples, the transfer functions of the equivalent spectrometer filters described in item 1 above can be changed from
\sin c
functions of frequency to other desired characteristics. This too is amenable to treatment by extension of the present results.