import numpy as np

def todo_spectrum(x):
    '''
    Input:
    x (N) - a waveform
    Output:
    spec (N) - magnitude spectrum at N frequencies between 0 and fs, not including fs.
    '''
    raise NotImplementedError('You need to write this part!')

def todo_findpeak(spec, fs, flo, fhi):
    '''
    Input:
    spec (N) - magnitude spectrum at N frequencies between 0 and fs, not including fs.
    fs (scalar)  - sampling frequency
    flo (scalar) -  low end of the frequency range to search
    fhi (scalar) - high end of the frequency range to search
    Output:
    fpeak (scalar) - frequency of the highest-magnitude spectral sample, in Hertz
    '''
    raise NotImplementedError('You need to write this part!')

def todo_zeros(freqs, fs):
    '''
    Input:
    freqs (nfreqs) - an array of frequencies you want zeroed out, in Hertz
    fs (scalar) - sampling frequency, in Hertz
    Output:
    r (2,nfreqs) - an array of complex zeros on the unit circle, in complex conjugate pairs
    '''
    raise NotImplementedError('You need to write this part!')

def todo_poles(r,BW,fs):
    '''
    Input: 
    r (2,nfreqs) - an array of complex zeros on the unit circle, in complex conjugate pairs
    BW (scalar) - desired bandwidth, in Hertz
    fs (scalar) - sampling frequency, in Hertz
    Output:
    p (2,nfreqs) - an array of complex poles with bandwidth BW, in complex conjugate pairs
    '''
    raise NotImplementedError('You need to write this part!')

def todo_coefficients(r):
    '''
    Input: 
    r (2,nfreqs) - an array of complex roots, in complex conjugate pairs
    Output:
    b (3,nfreqs) - an array of second-order polynomial coefficients, one per complex root pair
    '''
    raise NotImplementedError('You need to write this part!')

def todo_freqresponse(a, b, N):
    '''
    Input: 
    a (3) - feedback coefficients.  You may assume a[0]=1.
    b (3) - feedforward coefficients.  You may assume b[0]=1.
    N (scalar) - number of samples of the frequency response to compute
    Output: 
    omega (N) - frequencies linearly spaced between 0 and 2pi, not including 2pi.
    H (N) - B(e^{jw})/A(e^{jw}) evaluated at the frequencies in the vector omega.
    '''
    raise NotImplementedError('You need to write this part!')
    
def todo_filter(x, a, b):
    '''
    Input: 
    a (3) - feedback coefficients.  You may assume a[0]=1.
    b (3) - feedforward coefficients.  You may assume b[0]=1.
    x (N) - input waveform
    Output: 
    y (N) - output after being filtered using B(z)/A(z)
      Assume that x[n]==0 for n<0.
      Do not generate samples of y[n] for n >= N.
    '''
    raise NotImplementedError('You need to write this part!')

def todo_bell(sample_rate, frequencies, decay_times, duration):
    '''
    x = todo_bell(sample_rate, frequencies, decay_times, duration)
    Generate the sound of a bell ringing at a particular frequency.

    Input:
    sample_rate (integer) - sampling rate, in samples/second
    frequencies (nfreqs) - frequencies of the bell, in Hertz
    decay_times (nfreqs) - times after onset (in seconds) at which each tone decays to exp(-1)
    duration (float) - duration of x[n], in seconds
    
    Output:
    x (int(sample_rate*duration)) - the sound of the bell, given by
      x[n] = sum over k of (1/sin(omega[k])) exp(-sigma[k]*n) sin(omega[k]*(n+1))
    for 0 <= n <= int(sample_rate*duration), 
    and for omega[k] and sigma[k] determined from sample_rate, frequencies[k], and decay_times[k].
    '''
    raise NotImplementedError('You need to write this part!')
