**4. Efficiency of the partitioned time-frequency technique**

The effectiveness of the multitime ETHB technique is nowadays widely recognized by the RF and microwave community. The efficiency of the partitioned time-frequency simulation technique described in the previous section was also already established, as a consequence of the considerable reductions in the computational effort required to obtain the numerical solution of several RF circuits with distinct topologies and levels of complexity [16]. Even so, a brief comparison between this method, the previous state-of-the-art multitime ETHB and a conventional univariate time-step integration scheme (SPICE-like simulation), is included in this section. This will help the reader to get a perception of the potential of the partitioned hybrid technique. For that, we considered the RF mixer (frequency translation device) depicted in **Figure 3** as the illustrative application example. The circuit was simulated in MATLAB with three different techniques: (i) the partitioned time-frequency simulation technique, (ii) the multitime ETHB, and (iii) the Gear-2 multistep method [5] (a time-step integrator commonly used by SPICE-like commercial simulators).

Numerical computation times for simulations in the [0, 1.0 *μ*s] and [0, 10.0 *μ*s] intervals are presented in **Table 1**. For simplicity, in the hybrid time-frequency techniques we assumed a uniform grid in the *tE* slow time scale (we have chosen *hE* = 10 ns as the step size in that dimension) and we considered *K* = 11 as the maximum harmonic order for the HB evaluations in the *tC* fast carrier time scale.

By comparing the CPU times obtained with the methods, we can attest the superiority of the partitioned time-frequency method. Indeed, speedups of more than two times were obtained with this method in comparison to multitime ETHB. We can also attest the inefficiency of univariate time-step integration when dealing with RF problems. Finally, it must be noted that the efficiency gain of the partitioned time-frequency technique was achieved without com‐ promising accuracy. Indeed, the maximum discrepancy between solutions computed with this technique and multitime ETHB was in the order of 10−7 for all the state variables of the circuit.

**Figure 3.** Simplified schematic diagram of a mixer (frequency translation device).


**Table 1.** CPU time (h:min:sec)—simulation of the circuit depicted in **Figure 3**.
