On The Design of All-Pass Signals With Peak Amplitude Constraints

The problem of designing digital signals for testing (e.g., evaluating the impulse response) digital systems is one which has received very little attention in the digital signal-processing literature. This is because the impulse function is used as the standard test signal for most systems. Although the impulse function is suitable for this purpose in a wide variety of digital systems, there are cases in which the use of the impulse function leads to problems. Generally, such systems are those that have limited dynamic range--e.g., digital hardware implementations of a system, or fixed-point, finite, precision, software implementation of a digital system. In this paper, the problem is con395 sidered of designing signals other than the standard impulse function to be used to test digital systems of limited dynamic range. The desirable features of a test signal for digital systems are (i) It must be an all-pass signal in that it must be capable of testing the system (i.e., determining the frequency response of the system) for any admissible frequency. (it) It should be of limited duration. (m) It should be peak-amplitude-limited, to give the maximum utilization of the limited dynamic range of the system. The above features define a desirable test signal as one whose energy is spread out as much as possible to reduce the peak signal amplitude and therefore be able to pass higher energy signals through the system without clipping. If we let x{n) denote the test signal, then the requirements described above can be related to x(n) and X(eju), the Fourier transform of x{n), in the following manner.