| # This program is public domain |
| # Authors: Paul Kienzle, Nadav Horesh |
| """ |
| Chirp z-transform. |
| |
| CZT: callable (x,axis=-1)->array |
| define a chirp-z transform that can be applied to different signals |
| ZoomFFT: callable (x,axis=-1)->array |
| define a Fourier transform on a range of frequencies |
| ScaledFFT: callable (x,axis=-1)->array |
| define a limited frequency FFT |
| |
| czt: array |
| compute the chirp-z transform for a signal |
| zoomfft: array |
| compute the Fourier transform on a range of frequencies |
| scaledfft: array |
| compute a limited frequency FFT for a signal |
| """ |
| __all__ = ['czt', 'zoomfft', 'scaledfft'] |
| |
| import math, cmath |
| |
| import numpy as np |
| from numpy import pi, arange |
| from scipy.fftpack import fft, ifft, fftshift |
| |
| class CZT: |
| """ |
| Chirp-Z Transform. |
| |
| Transform to compute the frequency response around a spiral. |
| Objects of this class are callables which can compute the |
| chirp-z transform on their inputs. This object precalculates |
| constants for the given transform. |
| |
| If w does not lie on the unit circle, then the transform will be |
| around a spiral with exponentially increasing radius. Regardless, |
| angle will increase linearly. |
| |
| The chirp-z transform can be faster than an equivalent fft with |
| zero padding. Try it with your own array sizes to see. It is |
| theoretically faster for large prime fourier transforms, but not |
| in practice. |
| |
| The chirp-z transform is considerably less precise than the |
| equivalent zero-padded FFT, with differences on the order of 1e-11 |
| from the direct transform rather than the on the order of 1e-15 as |
| seen with zero-padding. |
| |
| See zoomfft for a friendlier interface to partial fft calculations. |
| """ |
| def __init__(self, n, m=None, w=1, a=1): |
| """ |
| Chirp-Z transform definition. |
| |
| Parameters: |
| ---------- |
| n: int |
| The size of the signal |
| m: int |
| The number of points desired. The default is the length of the input data. |
| a: complex |
| The starting point in the complex plane. The default is 1. |
| w: complex or float |
| If w is complex, it is the ratio between points in each step. |
| If w is float, it serves as a frequency scaling factor. for instance |
| when assigning w=0.5, the result FT will span half of frequncy range |
| (that fft would result) at half of the frequncy step size. |
| |
| Returns: |
| -------- |
| CZT: |
| callable object f(x,axis=-1) for computing the chirp-z transform on x |
| """ |
| if m is None: |
| m = n |
| if w is None: |
| w = cmath.exp(-1j*pi/m) |
| elif type(w) in (float, int): |
| w = cmath.exp(-1j*pi/m * w) |
| else: |
| w = cmath.sqrt(w) |
| self.w, self.a = w, a |
| self.m, self.n = m, n |
| |
| k = arange(max(m,n)) |
| wk2 = w**(k**2) |
| nfft = 2**nextpow2(n+m-1) |
| self._Awk2 = (a**-k * wk2)[:n] |
| self._nfft = nfft |
| self._Fwk2 = fft(1/np.hstack((wk2[n-1:0:-1], wk2[:m])), nfft) |
| self._wk2 = wk2[:m] |
| self._yidx = slice(n-1, n+m-1) |
| |
| def __call__(self, x, axis=-1): |
| """ |
| Parameters: |
| ---------- |
| x: array |
| The signal to transform. |
| axis: int |
| Array dimension to operate over. The default is the final |
| dimension. |
| |
| Returns: |
| ------- |
| An array of the same dimensions as x, but with the length of the |
| transformed axis set to m. Note that this is a view on a much |
| larger array. To save space, you may want to call it as |
| y = czt(x).copy() |
| """ |
| x = np.asarray(x) |
| if x.shape[axis] != self.n: |
| raise ValueError("CZT defined for length %d, not %d" % |
| (self.n, x.shape[axis])) |
| # Calculate transpose coordinates, to allow operation on any given axis |
| trnsp = np.arange(x.ndim) |
| trnsp[[axis, -1]] = [-1, axis] |
| x = x.transpose(*trnsp) |
| y = ifft(self._Fwk2 * fft(x*self._Awk2, self._nfft)) |
| y = y[..., self._yidx] * self._wk2 |
| return y.transpose(*trnsp) |
| |
| |
| def nextpow2(n): |
| """ |
| Return the smallest power of two greater than or equal to n. |
| """ |
| return int(math.ceil(math.log(n)/math.log(2))) |
| |
| |
| def ZoomFFT(n, f1, f2=None, m=None, Fs=2): |
| """ |
| Zoom FFT transform definition. |
| |
| Computes the Fourier transform for a set of equally spaced |
| frequencies. |
| |
| Parameters: |
| ---------- |
| n: int |
| size of the signal |
| m: int |
| size of the output |
| f1, f2: float |
| start and end frequencies; if f2 is not specified, use 0 to f1 |
| Fs: float |
| sampling frequency (default=2) |
| |
| Returns: |
| ------- |
| A CZT instance |
| A callable object f(x,axis=-1) for computing the zoom FFT on x. |
| |
| Sampling frequency is 1/dt, the time step between samples in the |
| signal x. The unit circle corresponds to frequencies from 0 up |
| to the sampling frequency. The default sampling frequency of 2 |
| means that f1,f2 values up to the Nyquist frequency are in the |
| range [0,1). For f1,f2 values expressed in radians, a sampling |
| frequency of 1/pi should be used. |
| |
| To graph the magnitude of the resulting transform, use:: |
| |
| plot(linspace(f1,f2,m), abs(zoomfft(x,f1,f2,m))). |
| |
| Use the zoomfft wrapper if you only need to compute one transform. |
| """ |
| if m is None: m = n |
| if f2 is None: f1, f2 = 0., f1 |
| w = cmath.exp(-2j * pi * (f2-f1) / ((m-1)*Fs)) |
| a = cmath.exp(2j * pi * f1/Fs) |
| return CZT(n, m=m, w=w, a=a) |
| |
| def ScaledFFT(n, m=None, scale=1.0): |
| """ |
| Scaled fft transform definition. |
| |
| Similar to fft, where the frequency range is scaled by a factor 'scale' and |
| divided into 'm-1' equal steps. Like the FFT, frequencies are arranged |
| from 0 to scale*Fs/2-delta followed by -scale*Fs/2 to -delta, where delta |
| is the step size scale*Fs/m for sampling frequence Fs. The intended use is in |
| a convolution of two signals, each has its own sampling step. |
| |
| This is equivalent to: |
| |
| fftshift(zoomfft(x, -scale, scale*(m-2.)/m, m=m)) |
| |
| For example: |
| |
| m,n = 10,len(x) |
| sf = ScaledFFT(n, m=m, scale=0.25) |
| X = fftshift(fft(x)) |
| W = linspace(-8, 8*(n-2.)/n, n) |
| SX = fftshift(sf(x)) |
| SW = linspace(-2, 2*(m-2.)/m, m) |
| plot(X,W,SX,SW) |
| |
| Parameters: |
| ---------- |
| n: int |
| Size of the signal |
| m: int |
| The size of the output. |
| Default: m=n |
| scale: float |
| Frequenct scaling factor. |
| Default: scale=1.0 |
| |
| Returns: |
| ------- |
| function |
| A callable f(x,axis=-1) for computing the scaled FFT on x. |
| """ |
| if m is None: |
| m = n |
| w = np.exp(-2j * pi / m * scale) |
| a = w**(m//2) |
| transform = CZT(n=n, m=m, a=a, w=w) |
| return lambda x, axis=-1: fftshift(transform(x, axis), axes=(axis,)) |
| |
| def scaledfft(x, m=None, scale=1.0, axis=-1): |
| """ |
| Partial with a frequency scaling. |
| See ScaledFFT doc for details |
| |
| Parameters: |
| ---------- |
| x: input array |
| m: int |
| The length of the output signal |
| scale: float |
| A frequency scaling factor |
| axis: int |
| The array dimension to operate over. The default is the |
| final dimension. |
| |
| Returns: |
| ------- |
| An array of the same rank of 'x', but with the size if |
| the 'axis' dimension set to 'm' |
| """ |
| return ScaledFFT(x.shape[axis], m, scale)(x,axis) |
| |
| def czt(x, m=None, w=1.0, a=1, axis=-1): |
| """ |
| Compute the frequency response around a spiral. |
| |
| Parameters: |
| ---------- |
| x: array |
| The set of data to transform. |
| m: int |
| The number of points desired. The default is the length of the input data. |
| a: complex |
| The starting point in the complex plane. The default is 1. |
| w: complex or float |
| If w is complex, it is the ratio between points in each step. |
| If w is float, it is the frequency step scale (relative to the |
| normal dft frquency step). |
| axis: int |
| Array dimension to operate over. The default is the final |
| dimension. |
| |
| Returns: |
| ------- |
| An array of the same dimensions as x, but with the length of the |
| transformed axis set to m. Note that this is a view on a much |
| larger array. To save space, you may want to call it as |
| y = ascontiguousarray(czt(x)) |
| |
| See zoomfft for a friendlier interface to partial fft calculations. |
| |
| If the transform needs to be repeated, use CZT to construct a |
| specialized transform function which can be reused without |
| recomputing constants. |
| """ |
| x = np.asarray(x) |
| transform = CZT(x.shape[axis], m=m, w=w, a=a) |
| return transform(x,axis=axis) |
| |
| def zoomfft(x, f1, f2=None, m=None, Fs=2, axis=-1): |
| """ |
| Compute the Fourier transform of x for frequencies in [f1, f2]. |
| |
| Parameters: |
| ---------- |
| m: int |
| The number of points to evaluate. The default is the length of x. |
| f1, f2: float |
| The frequency range. If f2 is not specified, the range 0-f1 is assumed. |
| Fs: float |
| The sampling frequency. With a sampling frequency of |
| 10kHz for example, the range f1 and f2 can be expressed in kHz. |
| The default sampling frequency is 2, so f1 and f2 should be |
| in the range 0,1 to keep the transform below the Nyquist |
| frequency. |
| x : array |
| The input signal. |
| axis: int |
| The array dimension the transform operates over. The default is the |
| final dimension. |
| |
| Returns: |
| ------- |
| array |
| The transformed signal. The fourier transform will be calculate |
| at the points f1, f1+df, f1+2df, ..., f2, where df=(f2-f1)/m. |
| |
| zoomfft(x,0,2-2./len(x)) is equivalent to fft(x). |
| |
| To graph the magnitude of the resulting transform, use:: |
| |
| plot(linspace(f1,f2,m), abs(zoomfit(x,f1,f2,m))). |
| |
| If the transform needs to be repeated, use ZoomFFT to construct a |
| specialized transform function which can be reused without |
| recomputing constants. |
| """ |
| x = np.asarray(x) |
| transform = ZoomFFT(x.shape[axis], f1, f2=f2, m=m, Fs=Fs) |
| return transform(x,axis=axis) |
| |
| |
| def _test1(x,show=False,plots=[1,2,3,4]): |
| norm = np.linalg.norm |
| |
| # Normal fft and zero-padded fft equivalent to 10x oversampling |
| over=10 |
| w = np.linspace(0,2-2./len(x),len(x)) |
| y = fft(x) |
| wover = np.linspace(0,2-2./(over*len(x)),over*len(x)) |
| yover = fft(x,over*len(x)) |
| |
| # Check that zoomfft is the equivalent of fft |
| y1 = zoomfft(x,0,2-2./len(y)) |
| |
| # Check that zoomfft with oversampling is equivalent to zero padding |
| y2 = zoomfft(x,0,2-2./len(yover), m=len(yover)) |
| |
| # Check that zoomfft works on a subrange |
| f1,f2 = w[3],w[6] |
| y3 = zoomfft(x,f1,f2,m=3*over+1) |
| w3 = np.linspace(f1,f2,len(y3)) |
| idx3 = slice(3*over,6*over+1) |
| |
| if not show: plots = [] |
| if plots != []: |
| import pylab |
| if 0 in plots: |
| pylab.figure(0) |
| pylab.plot(x) |
| pylab.ylabel('Intensity') |
| if 1 in plots: |
| pylab.figure(1) |
| pylab.subplot(311) |
| pylab.plot(w,abs(y),'o',w,abs(y1)) |
| pylab.legend(['fft','zoom']) |
| pylab.ylabel('Magnitude') |
| pylab.title('FFT equivalent') |
| pylab.subplot(312) |
| pylab.plot(w,np.angle(y),'o',w,np.angle(y1)) |
| pylab.legend(['fft','zoom']) |
| pylab.ylabel('Phase (radians)') |
| pylab.subplot(313) |
| pylab.plot(w,abs(y)-abs(y1)) #,w,np.angle(y)-np.angle(y1)) |
| #pylab.legend(['magnitude','phase']) |
| pylab.ylabel('Residuals') |
| if 2 in plots: |
| pylab.figure(2) |
| pylab.subplot(211) |
| pylab.plot(w,abs(y),'o',wover,abs(y2),wover,abs(yover)) |
| pylab.ylabel('Magnitude') |
| pylab.title('Oversampled FFT') |
| pylab.legend(['fft','zoom','pad']) |
| pylab.subplot(212) |
| pylab.plot(wover,abs(yover)-abs(y2), |
| w,abs(y)-abs(y2[0::over]),'o', |
| w,abs(y)-abs(yover[0::over]),'x') |
| pylab.legend(['pad-zoom','fft-zoom','fft-pad']) |
| pylab.ylabel('Residuals') |
| if 3 in plots: |
| pylab.figure(3) |
| ax1=pylab.subplot(211) |
| pylab.plot(w,abs(y),'o',w3,abs(y3),wover,abs(yover), |
| w[3:7],abs(y3[::over]),'x') |
| pylab.title('Zoomed FFT') |
| pylab.ylabel('Magnitude') |
| pylab.legend(['fft','zoom','pad']) |
| pylab.plot(w3,abs(y3),'x') |
| ax1.set_xlim(f1,f2) |
| ax2=pylab.subplot(212) |
| pylab.plot(wover[idx3],abs(yover[idx3])-abs(y3), |
| w[3:7],abs(y[3:7])-abs(y3[::over]),'o', |
| w[3:7],abs(y[3:7])-abs(yover[3*over:6*over+1:over]),'x') |
| pylab.legend(['pad-zoom','fft-zoom','fft-pad']) |
| ax2.set_xlim(f1,f2) |
| pylab.ylabel('Residuals') |
| if plots != []: |
| pylab.show() |
| |
| err = norm(y-y1)/norm(y) |
| #print "direct err %g"%err |
| assert err < 1e-10, "error for direct transform is %g"%(err,) |
| err = norm(yover-y2)/norm(yover) |
| #print "over err %g"%err |
| assert err < 1e-10, "error for oversampling is %g"%(err,) |
| err = norm(yover[idx3]-y3)/norm(yover[idx3]) |
| #print "range err %g"%err |
| assert err < 1e-10, "error for subrange is %g"%(err,) |
| |
| def _testscaled(x): |
| n = len(x) |
| norm = np.linalg.norm |
| assert norm(fft(x)-scaledfft(x)) < 1e-10 |
| assert norm(fftshift(fft(x))[n/4:3*n/4] - fftshift(scaledfft(x,scale=0.5,m=n/2))) < 1e-10 |
| |
| def test(demo=None,plots=[1,2,3]): |
| # 0: Gauss |
| t = np.linspace(-2,2,128) |
| x = np.exp(-t**2/0.01) |
| _test1(x, show=(demo==0), plots=plots) |
| |
| # 1: Linear |
| x=[1,2,3,4,5,6,7] |
| _test1(x, show=(demo==1), plots=plots) |
| |
| # Check near powers of two |
| _test1(range(126-31), show=False) |
| _test1(range(127-31), show=False) |
| _test1(range(128-31), show=False) |
| _test1(range(129-31), show=False) |
| _test1(range(130-31), show=False) |
| |
| # Check transform on n-D array input |
| x = np.reshape(np.arange(3*2*28),(3,2,28)) |
| y1 = zoomfft(x,0,2-2./28) |
| y2 = zoomfft(x[2,0,:],0,2-2./28) |
| err = np.linalg.norm(y2-y1[2,0]) |
| assert err < 1e-15, "error for n-D array is %g"%(err,) |
| |
| # 2: Random (not a test condition) |
| if demo==2: |
| x = np.random.rand(101) |
| _test1(x, show=True, plots=plots) |
| |
| # 3: Spikes |
| t=np.linspace(0,1,128) |
| x=np.sin(2*pi*t*5)+np.sin(2*pi*t*13) |
| _test1(x, show=(demo==3), plots=plots) |
| |
| # 4: Sines |
| x=np.zeros(100) |
| x[[1,5,21]]=1 |
| _test1(x, show=(demo==4), plots=plots) |
| |
| # 5: Sines plus complex component |
| x += 1j*np.linspace(0,0.5,x.shape[0]) |
| _test1(x, show=(demo==5), plots=plots) |
| |
| # 6: Scaled FFT on complex sines |
| x += 1j*np.linspace(0,0.5,x.shape[0]) |
| if demo == 6: |
| demo_scaledfft(x,0.25,200) |
| _testscaled(x) |
| |
| |
| def demo_scaledfft(v, scale, m): |
| import pylab |
| shift = pylab.fftshift |
| n = len(v) |
| x = pylab.linspace(-0.5, 0.5 - 1./n, n) |
| xz = pylab.linspace(-scale*0.5, scale*0.5*(m-2.)/m, m) |
| pylab.figure() |
| pylab.plot(x, shift(abs(fft(v))), label='fft') |
| pylab.plot(x, shift(abs(scaledfft(v))),'ro', label='x1 scaled fft') |
| pylab.plot(xz, abs(zoomfft(v, -scale, scale*(m-2.)/m, m=m)), |
| 'bo',label='zoomfft') |
| pylab.plot(xz, shift(abs(scaledfft(v, m=m, scale=scale))), |
| 'gx', label='x'+str(scale)+' scaled fft') |
| pylab.gca().set_yscale('log') |
| pylab.legend() |
| pylab.show() |
| |
| if __name__ == "__main__": |
| # Choose demo in [0,4] to show plot, or None for testing only |
| test(demo=None) |
| |