File:Multirate upsampling (interpolation) filter.svg

Original file (SVG file, nominally 900 × 630 pixels, file size: 43 KB)

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Summary

Description

English: Depiction of one dot product, resulting in one output sample (in green), computed by a multirate filter. This illustrates the formula

y[j+nL]=\sum _{k=0}^{K}x[n-k]\cdot h[j+kL],\ \ j=0,1,\ldots ,L-1,

at Upsampling by an integer factor, for the case L=4, n=9, j=3. Three (L-1) conceptual "inserted zeros" are depicted between each pair of input samples. Omitting them from the calculation is what distinguishes a multirate filter from a monorate filter.

Date

21 December 2019

Source

Own work

Author

Bob K

Permission
(Reusing this file)

I, the copyright holder of this work, hereby publish it under the following license:

	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. CC0falsefalse

SVG development

The source code of this SVG is invalid due to 5 errors.

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Gnu Octave source

click to expand

This graphic was created with the help of the following Octave script:

pkg load signal
graphics_toolkit gnuplot

 M = 1000;
 darkgreen = [33 150 33]/256;
 interpolation_factor = 4;

% Generate M+1 samples of a Gaussian window (filter type not important)
 filter = .3*exp(-.5*(((0:M)-M/2)/(.4*M/2)).^2);
% Sample the window.
 N=24;
 sam_per_hop = M/N;
 window_dots = filter(1+(0:N)*sam_per_hop);
 normalize   = sum(window_dots)/interpolation_factor;

 window_scale_factor = 0.5;
 sam_per_hop = sam_per_hop*window_scale_factor;
 
%------------------------------------------------------------------
 figure("position", [100 200 900 600])

% Plot the continuous filter function
 xoffset = sam_per_hop*15;
 yoffset = 0.1;
 
 plot(xoffset+(0:M)*window_scale_factor, yoffset+filter,  "linestyle",":", "linewidth",1, "color","black")
 set(gca, "xaxislocation", "origin")
 xlim([0 M])
 ylim([-.06 .45])                    % allows space for negative samples (in case I change the signal)
 set(gca, "ygrid","off");
 set(gca, "xgrid","on");
 set(gca, "ytick",[0], "fontsize",14);
 xticks = [0:4*sam_per_hop:M];
 set(gca, "xtick",xticks)
 set(gca,"xticklabel",[0:length(xticks)])

% Plot the filter coefficients
 hold on
 plot(xoffset+(0:N)*sam_per_hop, yoffset+window_dots,  "color","red", ".", "markersize",10)

% Create signal to be interpolated
 samples_per_cycle = 4*M;
 signal = .2*sin(2*pi*(0:M)/samples_per_cycle);

 signal_dots = signal(1:sam_per_hop:end);
% Simulate "inserted zeros", for display
 signal_dots(2:4:end) = 0;
 signal_dots(3:4:end) = 0;
 signal_dots(4:4:end) = 0;

% Plot the data
 L = length(signal_dots);
 plot((0:L-1)*sam_per_hop, signal_dots,  "color","blue", ".", "markersize",10)
 
% Compute dot product, and plot it
 dot_product = sum(window_dots(24:-4:4).*signal_dots(17:4:37))/normalize;
 x = 27*sam_per_hop;
 plot(x, dot_product, "color",darkgreen, ".", "markersize",14)
 plot([x,x],[0,.45]);                                   % vertical line
 
% xlabel('\leftarrow  n  \rightarrow', "fontsize",16)
 text(465, -.05, '\leftarrow  n  \rightarrow', "fontsize",16)
 text(18, .1, "X[n]", "fontsize",16, "color","blue")

 title("Multirate interpolation filter", "fontsize",16, "fontweight","normal");

File history

Click on a date/time to view the file as it appeared at that time.

	Date/Time	Dimensions	User	Comment
current	13:44, 24 December 2019	900 × 630 (43 KB)	Bob K	add a vertical line for filter delay arrow to point at
	15:12, 23 December 2019	900 × 630 (43 KB)	Bob K	replace random noise with more realistic signal
	15:54, 22 December 2019	855 × 585 (41 KB)	Bob K	Label y-origin. Indicate filter delay.
	14:43, 21 December 2019	855 × 585 (36 KB)	Bob K	User created page with UploadWizard