File:Frequency response of 3-term boxcar filter.svg

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English: Frequency response (discrete time Fourier transform) of a 3-tap boxcar filter. To produce these plots, samples of the DTFT are obtained by computing the DFT of the zero-filled impulse response. Alternatively, one can directly sample the closed form of the DTFT:   and plot the magnitude and angle of the samples. (See File:MA2Amp_C.svg)
Note: At the two frequencies where the magnitude goes to zero, there are phase discontinuities of π radians. They are not artifacts of phase wrapping. Rather, they indicate a sign inversion... multiplication by -1. Since that is equivalent to an amplitude modification, it does not affect the filter's property of phase linearity. An illustration of that can be viewed at File:Amplitude & phase vs frequency for 3-term boxcar filter.svg.
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Author original: Bob K
vector version: Krishnavedala
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(Reusing this file)
I, the copyright holder of this work, hereby publish it under the following license:
Creative Commons CC-Zero This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication.
The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.

Other versions This file was derived from: Frequency response of 3-term boxcar filter.gif
SVG development
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The SVG code is valid.
 
This vector image was created with Python by Krishnavedala.
Python source
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This script is a translation of the original Octave script into Python, for the purpose of generating an SVG file to replace the GIF version. 

import scipy
from scipy import signal
import numpy as np
from matplotlib import pyplot as plt
plt.rcParams['svg.fonttype'] = 'none'

N = 256
h = np.array([1., 1., 1.]) / 3
H = scipy.fftpack.shift(scipy.fft(h, N), np.pi)
w = np.linspace(-N/2, N/2-1, num=N) * 2 * np.pi / N

fig = plt.figure(figsize=[5,5])
plt.subplot(211)
plt.plot(w, np.abs(H), 'blue')
plt.grid(True)
plt.ylabel('Magnitude')
plt.xlim([-np.pi,np.pi])
plt.xticks([-np.pi, -2*np.pi/3,0,2*np.pi/3,np.pi], [])
plt.subplot(212)
plt.plot(w, np.angle(H), 'blue')
plt.grid(True)
plt.ylabel('Phase (radians)')
plt.xlabel('$\\longleftarrow$ Frequency ($\\omega$) (radians/sample) $\\longrightarrow$')
plt.xticks([-np.pi, -2*np.pi/3,0,2*np.pi/3,np.pi], ['-$\pi$','-2$\pi$/3','0','2$\pi$/3','$\pi$'])
plt.xlim([-np.pi,np.pi])
plt.yticks([-np.pi, -2,-1,0,1,2,np.pi], ['-$\pi$','-2','-1','0','1','2','$\pi$'])
plt.ylim([-np.pi,np.pi])
plt.subplots_adjust(hspace=0.1)
plt.savefig('Frequency response of 3-term boxcar filter.svg', bbox_inches='tight', transparent=True)
Octave/gnuplot source
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This script was derived from the original in order to address some GNUplot bugs: a missing title and two missing axis labels.  And to add an Octave print function, which creates an SVG file.  Alternatively, the gnuplot screen image has an export function that produces an SVG file, but the π characters aren't as professional-looking.  I think the resultant quality produced by this script is now better than the file produced by the Python script.
graphics_toolkit gnuplot
clear all; close all; clc
  
hfig = figure("position",[100 100 509 509]);

x1 = .12;               % left margin for name of Y-variable
x2 = .02;               % right margin
y1 = .10;               % bottom margin for ticks
y2 = .08;               % top margin for title
dy = .08;               % vertical space between rows

width = 1-x1-x2;
height= (1-y1-y2-dy)/2; % space allocated for each of 2 rows

x_origin = x1;
y_origin = 1;           % start at top of graph area
%=======================================================
  N= 256;
  h = [1 1 1]/3;              % impulse response
  H = fftshift(fft(h,N));     % samples of DTFT
  abscissa = (-N/2:N/2-1)*2*pi/N;  % normalized frequency
%=======================================================
  y_origin = y_origin -y2 -height;        % position of top row
  subplot("position",[x_origin y_origin width height])
  plot(abscissa, abs(H), "linewidth",2);
  xlim([-pi pi])
  ylim([0 1.2])
  set(gca, "XTick", [-pi -2*pi/3 0 2*pi/3 pi])
  set(gca, "YTick", [0 .2 .4 .6 .8 1])
  grid("on")

  ylabel("Magnitude")
% set(gca, "ticklabelinterpreter", "tex")	% tex is the default
  set(gca, "XTickLabel", ['-\pi'; '-2\pi/3'; '0'; '2\pi/3'; '\pi';])
  set(gca, "YTickLabel", ['0'; '.2'; '.4'; '.6'; '.8'; '1';])
  title("Frequency response of 3-term boxcar filter", "fontsize", 12)
%=======================================================
  y_origin = y_origin -dy -height;
  subplot("position",[x_origin y_origin width height])
  plot(abscissa, angle(H), "linewidth",2);
  xlim([-pi pi])
  ylim([-pi pi])
  set(gca, "XTick", [-pi -2*pi/3 0 2*pi/3 pi])
  set(gca, "YTick", [-pi -2 -1 0 1 2 pi])
  grid("on")

  xlabel('\leftarrow Frequency (\omega) (radians/sample) \rightarrow')
  ylabel("Phase (radians)")
% set(gca, "ticklabelinterpreter", "tex")	% tex is the default
  set(gca, "XTickLabel", ['-\pi'; '-2\pi/3'; '0'; '2\pi/3'; '\pi';])
  set(gca, "YTickLabel", ['-\pi'; '-2'; '-1'; '0'; '1'; '2'; '\pi';])

% The print function results in nicer-looking "pi" symbols
% than the export function on the GNUPlot figure toolbar.
print(hfig,"-dsvg", "-S509,509","-color", ...
'C:\Users\BobK\Frequency response of 3-term boxcar filter.svg')

File history

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Date/TimeThumbnailDimensionsUserComment
current13:07, 1 October 2020Thumbnail for version as of 13:07, 1 October 2020416 × 400 (20 KB)Krishnavedala (talk | contribs)Text-to-graph aspect ratio renders poorly in thumbnails with text unreadable.
01:36, 3 July 2019Thumbnail for version as of 01:36, 3 July 2019512 × 512 (37 KB)Bob K (talk | contribs)Increase graph "linewidth" to 2.
22:11, 2 July 2019Thumbnail for version as of 22:11, 2 July 2019512 × 512 (37 KB)Bob K (talk | contribs)Enlarge image. Add title. Improve rendering of "pi" symbols.
15:35, 22 August 2017Thumbnail for version as of 15:35, 22 August 2017416 × 400 (20 KB)Krishnavedala (talk | contribs)User created page with UploadWizard

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