In modern radio astronomy, one of the key technologies is to widen the frequency coverage of an instrument. The effects of higher-order modes on an instrument associated with wider bandwidths have been reported, which may degrade observation precision. It is important to quantify the higher-order propagation modes, though their power is too small to measure directly. Instead of the direct measurement of modes, we make an attempt to deduce them based on measurable radiation patterns. Assuming a linear system, whose radiated field is determined as a superposition of the mode coefficients in an instrument, we obtain a coefficient matrix connecting the modes and the radiated field and calculate the pseudo-inverse matrix. To investigate the accuracy of the proposed method, we demonstrate two cases with numerical simulations, axially-corrugated horn case and offset Cassegrain antenna case, and the effect of random errors on the precision. Both cases showed the deduced mode coefficients with a precision of 10e-6 with respect to the maximum mode amplitude and 10e-3 degrees in phase, respectively. The calculation errors were observed when the random errors were smaller than 0.01 percent of the maximum radiated field amplitude, which was a much lower level compared with measurement precision. The demonstrated method works independently of the details of a system. The method can quantify the propagation modes inside an instrument and will be applied to most of linear components and antennas, which leads to various applications such as diagnosis of feed alignment and higher-performance feed design.