This paper describes the application of principal component analysis to reduce the random noise present in the hyperspectral infrared observations. Within a set of spectral observations the number of components needed to characterize the atmosphere is far less than the number of wavelengths observed, typically by a factor between 50 and 70. The higher-order components, which mainly serve to characterize noise, can be eliminated along with the noise that they characterize. The results obtained depend on the variability of the selected sets of observations and on specific instrument characteristics such as spectral resolution and noise statistics. For a set of 10,000 Fourier transform spectrometer (FTS) simulated spectra, whose standard deviation is about 10% of the mean, we were able to obtain noise reduction factors between 5 and 8. Results obtained from real FTS, with standard deviation of about 10% of the mean, indicated practical noise reduction between 5 and 6. To avoid loss of information in the presence of highly deviant observations, it is necessary to use a conservative number of principal components higher than the optimum to maximum noise reduction. However, even then, noise reduction factors of 4 are still achievable.