Fourier upsamplng a spike: why is odd n so big ? 28 Oct 2025

0-spike-upsample.md

Fourier upsamplng a spike: why is odd n so big ?

Keywords: resample upsample interpolate frequency-domain Fourier scipy

The problem:

from scipy.signal import resample  # in Fourier space

for n in [100, 101]:
	x = spike( n )  # [100 0 0 ...]
	y = resample( x, 2*n )  # interpolate / upsample
	print the middle part 
	plot

n 100  y middle * 100: [ -8  0 5   0 -2   0  -2   0 5  0 -8]
n 101  y middle * 100: [  0 99 0 -99  0  99   0 -99 0 99  0]

spike is an extreme or silly testcase, but why is odd n so big ?

resample does: rfft( x ), 0-pad that at the end (inside irfft), irfft.

Links

resample doc
resample .py
wikipedia Upsampling interleaves 0s, [x0 0 x1 0 ...] in the time domain
dsp.stack why-to-pad-zeros-at-the-middle ff.

cheers
— denis 2025-10-28 October

spike-upsample.py

#!/usr/bin/env python3
r"""Spike [100 0 0 ...] $\ \to \ $ resample $ %d \times nx $
https://docs.scipy.org/doc/scipy/reference/generated/scipy.signal.resample.html
"""
# keywords: resample upsample interpolate frequency-domain Fourier scipy
# https://github.com/scipy/scipy/blob/main/scipy/signal/_signaltools.py
# $scipy/signal/_signaltools.py resample
# $scipy/fft/_basic.py irfft
# https://en.wikipedia.org/wiki/Upsampling#Upsampling_by_an_integer_factor
#   *separated by* L − 1 zeros

import numpy as np
from scipy.signal import resample
import matplotlib.pyplot as pl
import sys

print( 80 * "▄" )
np.set_printoptions( edgeitems=10, threshold=10, linewidth=120,
    formatter = dict( float = lambda x: "%.2g" % x ))  # 2 digits

def spike( n ):
    x = np.zeros( n )
    x[0] = 100
    return x

ns = [100, 101]
zoom = 2
plot = 0
    # to change these params, run this.py  a=1  b=2,None  'c = expr' ... in sh or ipython --
for arg in sys.argv[1:]:
    exec( arg )

title = __doc__.strip() % zoom
print( title, "\n" )
if plot:
    fig, axes = pl.subplots( nrows=len(ns), figsize=[8, 5] )
    fig.suptitle( title, multialignment="left" )
else:
    axes = len(ns) * [None]

for n, ax in zip( ns, axes ):
    x = spike( n )
    print( "\n" + 80 * "-" )
    nout = zoom * n
#...............................................................................
    y = resample( x, nout )
    jmid = nout // 2
    ymid = (y[jmid-5 : jmid+5+1] * 100) .round().astype(int)
    print( "n %d  y middle * 100: %s " % ( n, ymid ))
    if plot:
        ax.set_title( f"n = {n}", loc="center", y=0.9 )
        ax.set( ylim=[-2, 2] )
        ax.plot( y, "s-", lw=0.36, ms=1, color="crimson" )

if plot >= 2:
    import zz
    zz.savefig( __file__, info=__file__ )  # sign, date