  In this study, a novel type of Fourier transform radio spectrometer (termed
as all-digital radio spectrometer; ADRS) has been developed in which all
functionalities comprising a radio spectrometer including a sampler and Fourier
computing unit were implemented as a soft-core on a field-programmable gate
array (FPGA). A delay-line-based ramp-compare analog-to-digital converter
(ADC), one of completely digital ADC, was used, and two primary elements of the
ADC, an analog-to-time converter (ATC) and a time-to-digital converter (TDC),
were implemented on the FPGA. The sampling rate of the ADRS $f$ and the
quantization bit rate $n$ are limited by the relation, $\tau =
\frac{1}{2^{n}f}$, where $\tau$ is the latency of the delay element of the
delay-line. Given that the typical latency of the delay element implemented on
FPGAs is $\sim10$ ps, adoption of a low quantization bit rate, which satisfies
the requirements for radio astronomy, facilitates the realization of a high
sampling rate up to $\sim$100 GSa/s. In addition, as the proposed \ADRS does
not require a discrete ADC and can be implemented on mass-produced evaluation
boards, its fabrication cost is much lower than that of conventional
spectrometers. The ADRS prototype was fabricated with values of $f$ = 600 MSa/s
and $n$ = 6.6 using a PYNQ-Z1 evaluation board, with a $\tau$ of 16.7 ps. The
performance of the prototype, including its linearity and stability, was
measured, and a test observation was conducted using the Osaka Prefecture
University 1.85-m mm-submm telescope; this confirmed the potential application
of the prototype in authentic radio observations. With 10 times better cost
performance ($\sim$800 USD GHz$^{-1}$) than conventional radio spectrometers,
the prototype facilitates cost-effective coverage of intermediate frequency
(IF) bandwidths of $\sim100$ GHz in modern receiver systems.
