REALIZATION OF SHARP CUT‐OFF FREQUENCY CHARACTERISTICS ON DIGITAL COMPUTERS*

Sharp cut‐off frequency filtering is carried out in the discrete time domain on digital computers. A convolution of the digital filter impulse response with the sampled input yields the output. For practical reasons, the length of the filter inpulse response, corresponding to the number of filter coefficients, is limited, and consequently the resulting frequency characteristic will no longer be identical to that originally specified. This is analogous to synthesising some specified frequency characteristic with a finite number of resistive, capacitative and inductive components.

In Part I of this paper, we examine the effect of approximating the sharp cut‐off frequency characteristic best in a mean square sense by an impulse response of finite length. The resulting frequency characteristic corresponds to the truncated impulse response of the specified frequency characteristic. It has a cut‐off slope proportional to, and a mean square error inversely proportional to, the length of the impulse response, and is a biassed odd function about the cut‐off frequency point. Because of the Gibbs phenomenon for discontinuous functions, the resulting frequency characteristic will always have a maximum overshoot with respect to the specified characteristic of ± 9%, regardless of the length of the corresponding impulse response. Equal length truncated impulse responses of specified filters with different cut‐off frequencies yield frequency characteristics which are almost identical about their respective cut‐off points. Now on a log frequency scale (as against a linear frequency scale implied previously) such characteristics may be made almost identical about the respective cut‐off points by having the truncated impulse responses composed of an equal number of zero crossings. Results for the low‐pass filter are applicable to the high‐pass and band‐pass characteristics.

In the latter case, the mean square error is double that for a single slope characteristic (low‐pass or high‐pass) and the slopes at both edges of the passband are approximately equal in magnitude to the length of the impulse response (linear frequency scale).

Part II of this paper is concerned with reducing the ± 9% overshoot that results from the discontinuous nature of the sharp cut‐off frequency characteristic and which is not dependent on the length of the truncated impulse response. The reduction is achieved, at the expense of the steepness of cut‐off for the resulting frequency characteristic, by the use of functions which weight the truncated impulse response of the specified frequency characteristic. These functions are called apodising functions. Among other variables, the length of the truncated weighted impulse response will determine the amount of maximum overshoot since the effective frequency characteristic being approximated is no longer a discontinuous function. The digital realization of the finite length impulse responses of Parts I and II is discussed in Part III, together with the optimum partially specified digital filter approximation to the desired frequency characteristic.