Designing an M-Order Adaptive Echo Canceller

Design an M-order adaptive echo canceller using u(n) as system input and d(n) as desired system response, where d(n) contains echoes of u(n). Signals are sampled at 8kHz, and the echo canceller must eliminate echoes within 16ms.

To achieve this objective, we need to consider the following aspects:

1. Echo canceller structure and parameters: The echo canceller requires sufficient filter order to handle echo signals, ensuring echo elimination within 16ms. The filter order M should be calculated based on the sampling rate (8kHz) and required echo path coverage (16ms corresponds to 128 samples at 8kHz sampling). Key parameters include step size for adaptation and regularization factor for numerical stability.

2. Echo signal identification and separation: The echo canceller must analyze and process input and desired response signals to identify and separate echo components. This can be implemented using adaptive filtering techniques and correlation analysis. The Least Mean Squares (LMS) or Normalized LMS (NLMS) algorithm can be employed to continuously update filter coefficients based on the error signal e(n) = d(n) - y(n), where y(n) is the filter output.

3. Echo cancellation algorithms: Once echo signals are identified and separated, specific cancellation algorithms must be applied. Common implementations include adaptive filtering algorithms like LMS, NLMS, or RLS (Recursive Least Squares), which automatically adjust filter coefficients to minimize the mean square error. The algorithm implementation typically involves weight vector updates: w(n+1) = w(n) + μ·e(n)·u(n) for LMS, where μ is the step size parameter controlling convergence speed and stability.

Through proper design considerations and algorithm selection, we can implement an M-order adaptive echo canceller that effectively eliminates echoes within 16ms while improving overall system response quality. Code implementation would include initializing filter coefficients, processing samples in real-time, and continuously adapting to changing echo path characteristics.