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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Let's see if we can get Aaron's delay network to recognize two different patterns.  First, let's create the patterns.  For this simple test, we'll just use a 1Hz sine wave and a 0.5Hz sine wave for the two patterns.
Step1:  Now let's create a network that represents a rolling window in time (Aaron's "delay network").  The process determines what sort of pattern the network will be optimized for -- here we just go with white noise of a maximum of 3Hz.  theta determines how big the rolling window is -- here we use 0.5 seconds.
Step2:  Now we need to create the training data for decoding out of the rolling window.  Our patterns are larger than the rolling window, so to create our training data we will take our patterns, shift them, and cut them down to the right size.  In order to then give that to nengo, we also need to project from the window's space to the internal representation space (using the inv_basis).
The target array is the desired output value for each of the slices of the pattern in eval_points.  We'll use 1 for pattern1 and -1 for pattern2.
Step3:  Now we can create a connection optimized to do this decoding
Step4:  Let's try feeding in those two patterns and see what the response is
Step5:  It successfully detects the two frequencies, outputting 1 for the 1Hz pattern (pattern1) and -1 for the 0.5Hz (pattern2)!
Note that it has never observed a transition between frequencies, so it's somewhat reasonable that it's confused at the transitions.  This could be fixed by adding more training data that includes such transitions.
Now let's try intermediate frequencies that it has never seen before
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Python Code:
s_pattern = 2000                # number of data points in the pattern
t = np.arange(s_pattern)*0.001  # time points for the elements in the patter
pattern1 = np.sin(t*np.pi*2)
pattern2 = np.sin(0.5*t*np.pi*2)
plt.plot(t, pattern1, label='pattern1')
plt.plot(t, pattern2, label='pattern2')
plt.legend(loc='best')
plt.show()
Explanation: Let's see if we can get Aaron's delay network to recognize two different patterns.  First, let's create the patterns.  For this simple test, we'll just use a 1Hz sine wave and a 0.5Hz sine wave for the two patterns.
End of explanation
net = nengo.Network()
with net:
    process = nengo.processes.WhiteSignal(period=100., high=3., y0=0)
    rw = nengolib.networks.RollingWindow(theta=0.5, n_neurons=3000, process=process, neuron_type=nengo.LIFRate())
Explanation: Now let's create a network that represents a rolling window in time (Aaron's "delay network").  The process determines what sort of pattern the network will be optimized for -- here we just go with white noise of a maximum of 3Hz.  theta determines how big the rolling window is -- here we use 0.5 seconds.
End of explanation
s_window = 500
t_window = np.linspace(0, 1, s_window)
inv_basis = rw.inverse_basis(t_window)
    
eval_points = []
target = []
for i in range(s_pattern):
    eval_points.append(np.dot(inv_basis, np.roll(pattern1, i)[:s_window]))
    target.append([1])
    eval_points.append(np.dot(inv_basis, np.roll(pattern2, i)[:s_window]))
    target.append([-1])
Explanation: Now we need to create the training data for decoding out of the rolling window.  Our patterns are larger than the rolling window, so to create our training data we will take our patterns, shift them, and cut them down to the right size.  In order to then give that to nengo, we also need to project from the window's space to the internal representation space (using the inv_basis).
The target array is the desired output value for each of the slices of the pattern in eval_points.  We'll use 1 for pattern1 and -1 for pattern2.
End of explanation
with net:
    result = nengo.Node(None, size_in=1)
    nengo.Connection(rw.state, result,
                     eval_points=eval_points, scale_eval_points=False,
                     function=target, synapse=0.1)
Explanation: Now we can create a connection optimized to do this decoding
End of explanation
model = nengo.Network()
model.networks.append(net)
with model:
    freqs = [1, 0.5]
    def stim_func(t):
        freq = freqs[int(t/5) % len(freqs)]
        return np.sin(t*2*np.pi*freq)
    
    stim = nengo.Node(stim_func)
    nengo.Connection(stim, rw.input, synapse=None)
    
    p_result = nengo.Probe(result)
    p_stim = nengo.Probe(stim)
sim = nengo.Simulator(model)
sim.run(10)
plt.plot(sim.trange(), sim.data[p_stim], label='input')
plt.plot(sim.trange(), sim.data[p_result], label='output')
plt.legend(loc='best')
Explanation: Let's try feeding in those two patterns and see what the response is
End of explanation
model = nengo.Network()
model.networks.append(net)
with model:
    freqs = [1, 0.5, 0.75, 0.875, 0.625]
    def stim_func(t):
        freq = freqs[int(t/5) % len(freqs)]
        return np.sin(t*2*np.pi*freq)
    
    stim = nengo.Node(stim_func)
    nengo.Connection(stim, rw.input, synapse=None)
    
    p_result = nengo.Probe(result)
    p_stim = nengo.Probe(stim)
sim = nengo.Simulator(model)
sim.run(25)
plt.plot(sim.trange(), sim.data[p_stim], label='input')
plt.plot(sim.trange(), sim.data[p_result], label='output')
plt.legend(loc='best')
Explanation: It successfully detects the two frequencies, outputting 1 for the 1Hz pattern (pattern1) and -1 for the 0.5Hz (pattern2)!
Note that it has never observed a transition between frequencies, so it's somewhat reasonable that it's confused at the transitions.  This could be fixed by adding more training data that includes such transitions.
Now let's try intermediate frequencies that it has never seen before:
End of explanation
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