how to redefine atan2 function from -inf to inf in matlab?

Extending atan2 in MATLAB for a Full -∞ to +∞ Phase Range

Explore methods to redefine MATLAB's atan2 function to provide a continuous phase output from negative infinity to positive infinity, crucial for applications requiring unwrapped phase.

MATLAB's atan2(y, x) function is widely used to calculate the four-quadrant inverse tangent, returning an angle in the range of (-π, π] radians. While this range is suitable for many applications, certain scenarios, particularly in signal processing, control systems, or physics simulations, require a phase that extends continuously from negative infinity to positive infinity. This is often referred to as 'unwrapped phase' and is essential when tracking cumulative angular displacement or integrating phase changes over time.

Understanding the Limitation of Standard atan2

The standard atan2 function inherently provides a principal value, meaning it always maps the input (y, x) pair to an angle within a single 2π interval. This behavior is mathematically sound for determining the angle of a vector in a Cartesian plane. However, when dealing with sequences of angles that might cross the ±π boundary multiple times, this 'wrapping' can lead to discontinuities. For instance, if an angle smoothly increases from π/2 to 3π/2, atan2 would report a jump from π/2 to -π/2, rather than a continuous increase.

flowchart TD
    A[Input: (y, x) pair] --> B{Calculate atan2(y, x)}
    B --> C[Output: Angle in (-π, π]]
    C --> D{Is continuous phase needed?}
    D -- No --> E[Use standard atan2 result]
    D -- Yes --> F[Apply phase unwrapping algorithm]
    F --> G[Output: Angle in (-∞, ∞)]

Flowchart illustrating the decision process for using standard vs. unwrapped atan2.

Implementing Phase Unwrapping for atan2

To extend the range of atan2 to (-∞, ∞), you need to implement a phase unwrapping algorithm. This typically involves tracking the phase changes between consecutive samples and adding or subtracting multiples of 2π whenever a 2π jump (or wrap) is detected. MATLAB provides the unwrap function, which is specifically designed for this purpose. It operates on an array of phase values, correcting the discontinuities.

% Example: Generating a complex signal with increasing phase
t = 0:0.01:10;
frequency = 1;
phase_true = 2 * pi * frequency * t; % True continuous phase

% Create a complex signal
complex_signal = exp(1i * phase_true);

% Calculate phase using atan2 (wrapped phase)
wrapped_phase = atan2(imag(complex_signal), real(complex_signal));

% Unwrap the phase
unwrapped_phase = unwrap(wrapped_phase);

% Plotting to visualize
figure;
plot(t, phase_true, 'b', 'DisplayName', 'True Phase');
hold on;
plot(t, wrapped_phase, 'r--', 'DisplayName', 'Wrapped Phase (atan2)');
plot(t, unwrapped_phase, 'g:', 'DisplayName', 'Unwrapped Phase');
xlabel('Time (s)');
ylabel('Phase (radians)');
title('Phase Unwrapping with atan2');
legend('Location', 'best');
grid on;

MATLAB code demonstrating the use of unwrap on atan2 output.

Manual Unwrapping for Custom Scenarios

While MATLAB's unwrap function is robust, there might be specific cases where you need more control or are working with non-sequential data. In such situations, you can implement a manual unwrapping logic. This typically involves iterating through the phase array and adjusting each point based on the previous point's value and the 2π boundary.

function unwrapped_phase = manual_unwrap(wrapped_phase)
    unwrapped_phase = zeros(size(wrapped_phase));
    unwrapped_phase(1) = wrapped_phase(1);
    for k = 2:length(wrapped_phase)
        diff = wrapped_phase(k) - wrapped_phase(k-1);
        if diff > pi
            unwrapped_phase(k) = wrapped_phase(k) - 2*pi;
        elseif diff < -pi
            unwrapped_phase(k) = wrapped_phase(k) + 2*pi;
        else
            unwrapped_phase(k) = wrapped_phase(k);
        end
        % Add the accumulated 2*pi shifts from previous steps
        unwrapped_phase(k) = unwrapped_phase(k) + (unwrapped_phase(k-1) - (wrapped_phase(k-1) - mod(wrapped_phase(k-1) + pi, 2*pi) + pi));
    end
end

% Example usage with the previous wrapped_phase
% unwrapped_phase_manual = manual_unwrap(wrapped_phase);
% figure;
% plot(t, phase_true, 'b', 'DisplayName', 'True Phase');
% hold on;
% plot(t, wrapped_phase, 'r--', 'DisplayName', 'Wrapped Phase (atan2)');
% plot(t, unwrapped_phase_manual, 'm-.', 'DisplayName', 'Manual Unwrapped Phase');
% xlabel('Time (s)');
% ylabel('Phase (radians)');
% title('Manual Phase Unwrapping');
% legend('Location', 'best');
% grid on;

A basic manual phase unwrapping function. Note: MATLAB's unwrap is generally more robust.

1. Generate Complex Data

Create a complex signal z = x + iy where x and y are the real and imaginary components, respectively. This could be from an FFT, a filter output, or a simulated signal.

2. Calculate Wrapped Phase

Use atan2(imag(z), real(z)) to obtain the initial wrapped phase values. This will be in the (-π, π] range.

3. Apply Unwrapping Function

Pass the wrapped_phase array to MATLAB's unwrap function: unwrapped_phase = unwrap(wrapped_phase). This will correct the 2π discontinuities.

4. Verify Results

Plot both the original wrapped_phase and the unwrapped_phase to visually confirm that the discontinuities have been removed and the phase now extends continuously.