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## PyTorch `torch.polar` Function `torch.polar` is a PyTorch function used to construct a complex tensor from its polar coordinates: absolute values (magnitudes) and angles (phases). --- ## Introduction In mathematics, a complex number can be represented in polar form as: $$z = r \cdot e^{i\theta} = r \cdot (\cos\theta + i\sin\theta)$$ Where: * $r$ is the absolute value (magnitude or modulus). * $\theta$ is the angle (phase or argument) in radians. `torch.polar` takes these two components as input tensors and returns a complex tensor representing $z$. This is particularly useful in fields like Digital Signal Processing (DSP), Fourier analysis, and quantum computing simulations. --- ## Syntax and Parameters ### Syntax ```python torch.polar(abs, angle, *, out=None) -> Tensor ``` ### Parameters | Parameter | Type | Description | | :--- | :--- | :--- | | **`abs`** | `Tensor` | The absolute value (magnitude) of the complex number. Must contain real values. | | **`angle`** | `Tensor` | The angle (argument) of the complex number in radians. Must contain real values. | | **`out`** (Optional) | `Tensor` | The output tensor. | ### Data Type Constraints * `abs` and `angle` must have the same shape. * `abs` and `angle` must be of float types (`torch.float32` or `torch.float64`). * If the inputs are `torch.float32`, the returned tensor will be `torch.complex64`. * If the inputs are `torch.float64`, the returned tensor will be `torch.complex128`. --- ## Code Examples ### Basic Example The following example demonstrates how to create a 1D complex tensor from magnitude and phase tensors. ```python import torch # Create the absolute value (magnitude) tensor abs_val = torch.tensor([1.0, 2.0, 3.0]) # Create the angle (radians) tensor angle = torch.tensor([0.0, torch.pi / 2, torch.pi]) # Construct the complex tensor from polar coordinates x = torch.polar(abs_val, angle) print("Constructed Complex Tensor:") print(x) # Verify by extracting the magnitude and angle back print("\nExtracted Absolute Values (Magnitudes):", x.abs()) print("Extracted Angles (Radians):", x.angle()) ``` ### Output ```text Constructed Complex Tensor: tensor([ 1.0000+0.0000j, 0.0000+2.0000j, -3.0000+0.0000j]) Extracted Absolute Values (Magnitudes): tensor([1., 2., 3.]) Extracted Angles (Radians): tensor([0.0000, 1.5708, 3.1416]) ``` --- ## Key Considerations 1. **Device Matching**: Both `abs` and `angle` must reside on the same device (e.g., both on CPU or both on the same GPU). 2. **Type Matching**: Both input tensors must have the same data type (e.g., both `torch.float32`). Mixing `float32` and `float64` will result in a runtime error. 3. **Inverse Operation**: You can perform the inverse operation (converting a complex tensor back to polar coordinates) using `torch.abs(x)` and `torch.angle(x)`.