The spectrum of clipped noise (1966)

The present report calculates in some detail the (intensity) spectrum to be expected for clipped (also called "limited") noise. Two cases are considered: (A), the clipping of an unmodulated noise band (DINA) [1] and (B) a carrier modulated by clipped noise. The computations are made for various shapes of noise bands before clipping, viz., 1) a uniform or rectangular structure, 2) a Gaussian distribution, 3) an "optical" (Lorentzian) shape factor of the type 1/[(v-v <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> ) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> + δ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ]. The simplest type of calculation to make is that for what we term "extreme clipping," wherein the limiting amplitude is very small compared to the rms amplitude before clipping. The mathematical theory for this is given in Section III, while Section IV develops the theory for clipping at an arbitrary level. The basic mathematical method, which is rather general and is useful, we believe, for a variety of noise problems, is presented in Section II and consists in utilizing a relation between the correlation function and the normal surface, along lines suggested by Rice [1]. The results of the calculation are discussed in Section I and are displayed in Figures 4-10 and Tables I and II. If the clipping is not down to more than the rms level before limiting (equivalent to clipping at about 1.4 times the rms level after clipping), there is practically no distortion of the spectrum. Even in the case of extreme clipping the wastage of power due to spoiling of the spectrum's uniformity is small, amounting to only 31 percent in (A) and 24 percent in (B). Of the 31 percent loss in (A), 19 percent is due to production of harmonics of the central frequency. Corresponding harmonics are absent in (B). Clipping is beneficial for jamming purposes in either (A) or (B) since it reduces the peak power requirements. In addition, in (B) it materially diminishes the wastage of power in the carrier frequency. These facts are demonstrated particularly clearly by Tables I and II. For instance, Table II shows us that in (B) the ratio of the energy in the noise sideband to that in the carrier is only 0.23 when the clipping level is twice the original rms noise level, but increases to 0.52 when these two levels are the same, and to 1.0 for extreme clipping. It is to be cautioned that the present report calculates only the (intensity) spectrum of the clipped noise, and does not deal with its effectiveness on a receiver, which we hope to discuss later from a quantitative standpoint [2]. We can, however, say qualitatively that if the receiver breadth is very small compared to the noise band, the received disturbance will have the same type of Gaussian fluctuation, and hence the same effectiveness as unclipped noise with the same spectral distribution. On the other hand, if the receiver is comparable with the noise band in width, there will be, due to the clipping, a tendency for a "ceiling" in the resultant deflection of the recording device, and under these conditions the utility of clipped noise for jamming is materially diminished.