The minimum mean-squared-error decision-feedback equalizer (MMSE-DFE) has properties that suggest that it is a canonical equalization structure for systems that combine equalization with coded modulation. The structure and performance of the MMSE-DFE are succinctly derived using linear-estimation-theoretic principles in this first part of this two-part paper. The front-end of the MMSE-DFE, called the "mean-square whitened matched filter" (MS-WMF), is preferable in some ways to a matched filter or a whitened matched filter as a canonical receiver front end. In a coded system, the feedback filter of the MMSE-DFE may be implemented in the transmitter using precoding. The MMSE-DFE can perform significantly better than a zero-forcing decision-feedback equalizer, particularly at moderate-to-low SNR's and on severe-ISI channels. The MMSE-DFE is biased. The optimum unbiased MMSE-DFE is the MMSE-DFE with the bias removed. Removing bias improves error probability, but reduces the SNR to SNR/sub MMSE-DFE,U/=SNR/sub MMSE-DFE/-1. It is shown that this SNR relationship is a particular case of a very general result and that SNR/sub MMSE-DFE,U/ gives a more realistic estimate of SNR. The results are extended to partial response equalization and to equalization with correlated inputs in an appendix.< >