The fir (finite impulse response) filter is an essential tool for a large number of applications in communication. In this paper we consider the design of linear‐phase fir digital filters with finitely precise coefficients. Coefficient inaccuracy is known to degrade the frequency response of band‐select fir filters, especially in the stopband region. We derive a bound on the attainable stopband attenuation, and we also develop techniques for designing fir filters with finitely precise coefficients. Mixed‐integer programming algorithms are presented to select finitely precise coefficients for a filter that best approximates an arbitrary magnitude characteristic in the minimax sense. Our method generates a number of possible solutions including that of simple rounding or truncation and then selects the best finitely precise coefficients from this set. In this way, significant improvement in the filter performance is gained over methods that simply round or truncate the infinitely precise coefficients. We also show how integer programming can be used to design filters with powers of two coefficients. Such filters are easier to mechanize since they do not require multipliers.
We present the constructive design of finite order equalizer filters for data transmission systems employing decision feedback equalization. Both transmitter design with power constraints and receiver design with ambient noise considerations are treated. Expressions for the filter tap settings which maximize a signal‐to‐noise ratio are found for both baseband pulse amplitude modulation and quadrature amplitude modulation (QAM) systems. Design examples are given in a passband equivalent (of QAM) formulation for an average toll telephone connection. Neglecting the possibility of error propagation, these examples demonstrate that decision feedback equalization requires fewer taps for acceptable system performance as compared to linear equalization. The problem of postcursor size in a decision feedback equalized response is treated and shown to diminish in importance when a hybrid equalization procedure is imposed on the linear tap adjustment. The price one pays for allowing the linear filter taps to reduce the postcursor sizes in this hybrid equalizer is a lower signal‐to‐noise ratio.
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