Numpy "double"-broadcasting - is it possible?

Question

Is it possible to use "double"-broadcasting to remove the loop in the following code? In other words, to broadcast across the entire time array T as well as the same-dimensioned arrays freqs and phases.

freqs = np.arange(100)
phases = np.random.randn(len(freqs))
T = np.arange(0, 500)

signal = np.zeros(len(T))
for i in xrange(len(signal)):
    signal[i] = np.sum(np.cos(freqs*T[i] + phases))

Answer 1

You can reshape T as a 2d array by adding a new axis to it, which will trigger the broadcasting when multiplied/added with a 1d array, and then later on use numpy.sum to collapse this axis:

np.sum(np.cos(freqs * T[:,None] + phases), axis=1)
#                      add new axis        remove it with sum

---

Testing:

(np.sum(np.cos(freqs * T[:,None] + phases), axis=1) == signal).all()
# True

​

Answer 2

One idea that just came to me (but which may be computationally expensive?) is to construct the arguments as a matrix:

phases = phases.reshape((len(phases), 1))
argumentMatrix = np.outer(freqs, T) + phases
cosineMatrix = np.cos(argumentMatrix)
signal = np.sum(cosineMatrix, axis=0) # sum, collapsing columns