High-speed high-resolution Analog-to-Digital Conversion is the key part for waveform digitization in physics experiments and many other domains. This paper presents a new fully digital correction of mismatch errors among the channels in Time Interleaved Analog-to-Digital Converter (TIADC) systems. We focus on correction with wide-band input signal, which means that we can correct the mismatch errors for any frequency point in a broad band with only one set of filter coefficients. Studies were also made to show how to apply the correction algorithm beyond the base band, i.e. other Nyquist zones in the under-sampling situation. Structure of the correction algorithm is presented in this paper, as well as simulation results. To evaluate the correction performance, we actually conducted a series of tests with two TIADC systems. The results indicate that the performance of both two TIADC systems can be greatly improved by correction, and the Effective Number Of Bits (ENOB) is successfully improved to be better than 9.5 bits and 5.5 bits for an input signal up to the bandwidth (-3dB) range in the 1.6-Gsps 14-bit and the 10-Gsps 8-bit TIADC systems, respectively. Tests were also conducted for input signal frequencies in the second Nyquist zone, which shows that the correction algorithms also work well as expected.