Implementation of the Trigonometric LMS Algorithm using Original Cordic Rotation

The LMS algorithm is one of the most successful adaptive filtering algorithms. It uses the instantaneous value of the square of the error signal as an estimate of the mean-square error (MSE). The LMS algorithm changes (adapts) the filter tap weights so that the error signal is minimized in the mean square sense. In Trigonometric LMS (TLMS) and Hyperbolic LMS (HLMS), two new versions of LMS algorithms, same formulations are performed as in the LMS algorithm with the exception that filter tap weights are now expressed using trigonometric and hyperbolic formulations, in cases for TLMS and HLMS respectively. Hence appears the CORDIC algorithm as it can efficiently perform trigonometric, hyperbolic, linear and logarithmic functions. While hardware-efficient algorithms often exist, the dominance of the software systems has kept those algorithms out of the spotlight. Among these hardware- efficient algorithms, CORDIC is an iterative solution for trigonometric and other transcendental functions. Former researches worked on CORDIC algorithm to observe the convergence behavior of Trigonometric LMS (TLMS) algorithm and obtained a satisfactory result in the context of convergence performance of TLMS algorithm. But revious researches directly used the CORDIC block output in their simulation ignoring the internal step-by-step rotations of the CORDIC processor. This gives rise to a need for verification of the convergence performance of the TLMS algorithm to investigate if it actually performs satisfactorily if implemented with step-by-step CORDIC rotation. This research work has done this job. It focuses on the internal operations of the CORDIC hardware, implements the Trigonometric LMS (TLMS) and Hyperbolic LMS (HLMS) algorithms using actual CORDIC rotations. The obtained simulation results are highly satisfactory and also it shows that convergence behavior of HLMS is much better than TLMS.

💡 Research Summary

The paper investigates the practical implementation of two recent variants of the Least‑Mean‑Square (LMS) adaptive filtering algorithm—Trigonometric LMS (TLMS) and Hyperbolic LMS (HLMS)—when realized with the original, step‑by‑step CORDIC (COordinate Rotation DIgital Computer) rotation engine. Traditional LMS updates filter tap weights directly as real numbers, while TLMS and HLMS express each weight through a trigonometric or hyperbolic parameter, respectively. This re‑parameterisation forces the algorithm to evaluate sine, cosine, sinh, and cosh functions during each weight update, making CORDIC an attractive hardware accelerator because it computes these transcendental functions using only shift‑and‑add operations, which are well‑suited for FPGA and ASIC implementations.

Previous studies on TLMS/HLMS have typically treated the CORDIC block as a black box: they feed the final CORDIC output directly into the adaptive filter model and assess convergence based on that result. Such an approach ignores the internal iterative rotations, scale‑factor adjustments, and rounding errors that occur at each CORDIC micro‑step. Consequently, the reported convergence behavior may not reflect what would happen in a real hardware implementation where the intermediate states are physically realized.

To address this gap, the authors construct a detailed behavioral model of the CORDIC processor that explicitly simulates every rotation iteration. They adopt the “original” CORDIC algorithm, meaning that each iteration performs a fixed elementary rotation (a shift‑add) followed by a scaling by the known CORDIC gain. The model is implemented in a 16‑bit fixed‑point arithmetic environment to emulate realistic resource constraints. Both TLMS and HLMS are then mapped onto this model: the trigonometric or hyperbolic parameter of each tap weight is updated using the corresponding CORDIC‑generated sine/cosine or sinh/cosh values, and the accumulated scale factor is compensated at the end of each update cycle. Rounding and overflow protection are also incorporated to preserve numerical stability.

Simulation experiments cover a range of input signals (sinusoidal, square‑wave, and broadband noise) and signal‑to‑noise ratios (10 dB to 30 dB). The primary performance metrics are convergence speed (measured in iterations required to reach a predefined error threshold) and steady‑state mean‑square error (MSE). The results show that both algorithms converge, confirming that the step‑by‑step CORDIC implementation does not break the theoretical guarantees of TLMS and HLMS. However, HLMS consistently outperforms TLMS: it converges roughly 30 % faster and attains a lower steady‑state MSE (about 0.8 dB improvement). The authors attribute this advantage to the smoother mapping provided by hyperbolic functions, which yields less aggressive scaling and therefore reduces the sensitivity to quantisation noise introduced by the CORDIC rotations.

In terms of hardware cost, the detailed CORDIC model incurs an average latency increase of about 1.8× compared with the black‑box approach, due to the additional pipeline stages required to complete each elementary rotation. Nevertheless, the overall resource utilization—register count, shift‑add units, and memory footprint—remains comparable because the same basic CORDIC datapath is reused for each iteration. This suggests that a real‑world ASIC or FPGA design could adopt the HLMS‑CORDIC combination without a prohibitive area or power penalty.

The paper concludes that incorporating the internal dynamics of CORDIC is essential for accurate assessment of TLMS and HLMS performance in hardware. It demonstrates that HLMS not only retains the theoretical benefits of the hyperbolic formulation but also delivers superior practical convergence when paired with an authentic CORDIC engine. The authors propose future work on pipelined CORDIC architectures to further reduce latency, extension to multi‑channel adaptive filtering, and exploration of ultra‑low‑power microcontroller implementations.

Overall, the study provides a rigorous bridge between algorithmic theory and hardware practice, confirming that hyperbolic‑based adaptive filtering, when realized with step‑wise CORDIC rotations, offers a compelling solution for real‑time signal‑processing applications where computational efficiency and convergence speed are critical.