An entirely new set of criteria for the design of kernels (generating functions) for time-frequency representations (TFRs) is presented. These criteria aim only to produce kernels (and thus, TFRs) which will enable more accurate classification. We refer to these kernels, which are optimized to discriminate among several classes of signals, as signal class dependent kernels, or simply class dependent kernels. The genesis of the class dependent kernel is to be found in the area of operator theory, which we use to establish a direct link between a discrete-time, discrete-frequency TFR and its corresponding discrete signal. We see that many similarities, but also some important differences, exist between the results of the continuous-time operator approach and our discrete one. The differences between the continuous representations and discrete ones may not be the simple sampling relationship which has often been assumed. From this work, we obtain a very concise, matrix-based expression for a discrete- time/discrete-frequency TFR which is simply the product of the kernel with another matrix. This simple expression opens up the possibility to optimize the kernel in a number of ways. We focus, of course, on optimizations most suitable for classification, and ultimately wind up with the class dependent kernel. When applied to simulated sonar transient signals, we find that our approach does a good job of discriminating within very similar classes of transients and is especially sensitive to differences in time variation across classes.