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{\rtf\ansi \deff0{\colortbl;\red0\green0\blue255;}{\fonttbl{\f0
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\pard File Name: 428ca4.mcd\par ECE 428 Computer Assignment #4\par Dr. 
James S. Kang, ECE Department, Cal Poly Pomona\par \par Consider a 
continuous-time signal x(t) = 2 cos(2{\f1 p}t).\par \par (a) Find the 
Fourier transform X(f) and plot it.\par \par (b) The signal x(t) is 
sampled at t = kTs where Ts = 0.5 second to get x1(k).  Find and plot 
the\par DFT of x1(k) for N = 2, 3, 4, 5, 6, 7, 8.  Compare the result 
with X(f) and interpret the result.\par \par \pard (c) The signal x(t) 
is sampled at t = kTs where Ts = 0.25 second to get x2(k).  Find and 
plot the\par DFT of x1(k) for N = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.  
Compare the result with X(f) and interpret the result.\par \par \pard 
(d) The signal x(t) is sampled at t = kTs where Ts = 0.125 second to 
get x3(k).  Find and plot the\par DFT of x1(k) for N = 2, 3, 4, 5, 6, 
7, 8, 9, 10, 11, 12, 16.  Compare the result with X(f) and interpret 
the result.}
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{\rtf\ansi \deff0{\colortbl;\red0\green0\blue255;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard When N is even, we have 
a nonzero spectrum at n = N/2 which is equivalent to fs*n/N Hz\par in 
analog frequency.  For example, for N = 8, 2*4/8 = 1 Hz.  The value of 
Ts*X(N/2)/N = 1.\par For example, for N = 8, (1/2)*16/8 = 1, which is 
the spectrum at f = 1 Hz.\par When N is odd, the additional sample 
causes the DFT to be different than the desired result.\par The 
spectrums are symmetrical with respect to N/2.  The spectrums with the 
highest magnitude\par are at n = (N-1)/2 and n = (N+1)/2.}
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.TXT 39 -32 489 0 0
Cg a71.000000,71.000000,362
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue255;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard For N = 4, 8, 12, the 
nonzero values occur at n = N/4 and n = 3N/4.  The corresponding \par 
analog frequencies are (fs)n/N.  For example, for N = 12, (fs)(N/4)/N 
= (4) (12/4)/12 = 1Hz,\par (fs)(3N/4)/N = (4)(3*12/4)/12 = 3Hz.  The 
values at n = N/4 and n = 3N/4 can be normalized\par by dividing by N.  
For example, for N = 12, 12/12 = 1 at n = 12/4 = 3 and n = 3*12/4 = 9. }
.TXT 10 -1 491 0 0
Cg a72.000000,72.000000,3
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue255;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard (d)}
.EQN 1 5 516 0 0
{0:Ts}NAME:0.125
.EQN 0 9 517 0 0
{0:k}NAME:0;16
.EQN 0 9 518 0 0
{0:x3}NAME({0:k}NAME):2*{0:cos}NAME(2*{0:\p}NAME*{0:k}NAME*{0:Ts}NAME)
.EQN 6 -22 519 0 0
{0:Xa8}NAME({0:n}NAME,{0:N}NAME):((0,{0:N}NAME-1,{0:k}NAME,{0:x3}NAME({0:k}NAME)*({0:e}NAME)^(-1j*{0:n}NAME*(2*{0:\p}NAME)/({0:N}NAME)*{0:k}NAME)){64})
.EQN 6 0 520 0 0
{0:N}NAME:2
.EQN 0 7 521 0 0
{0:n}NAME:0;{0:N}NAME-1
.EQN 3 -7 522 0 0
{0:Xa8}NAME({0:n}NAME,{0:N}NAME)=
.EQN 10 0 523 0 0
{0:N}NAME:3
.EQN 0 7 524 0 0
{0:n}NAME:0;{0:N}NAME-1
.EQN 3 -7 525 0 0
{0:Xa8}NAME({0:n}NAME,{0:N}NAME)=
.EQN 22 0 526 0 0
{0:N}NAME:4
.EQN 0 7 527 0 0
{0:n}NAME:0;{0:N}NAME-1
.EQN 2 23 667 0 0
&&(_n_u_l_l_&_n_u_l_l_)&|({0:Xa8}NAME({0:n}NAME,4)),0@&&(_n_u_l_l_&_n_u_l_l_)&{0:n}NAME
0 0 1 1 1 0 0 1 1 
0 0 1 1 1 0 0 1 1 
0 1 0 2 1 1 NO-TRACE-STRING
0 1 1 2 1 1 NO-TRACE-STRING
0 3 2 0 1 1 NO-TRACE-STRING
0 4 3 0 1 1 NO-TRACE-STRING
0 1 4 0 1 1 NO-TRACE-STRING
0 2 5 0 1 1 NO-TRACE-STRING
0 3 6 0 1 1 NO-TRACE-STRING
0 4 0 0 1 1 NO-TRACE-STRING
0 1 1 0 1 1 NO-TRACE-STRING
0 2 2 0 1 1 NO-TRACE-STRING
0 3 3 0 1 1 NO-TRACE-STRING
0 4 4 0 1 1 NO-TRACE-STRING
0 1 5 0 1 1 NO-TRACE-STRING
0 2 6 0 1 1 NO-TRACE-STRING
0 3 0 0 1 1 NO-TRACE-STRING
0 4 1 0 1 1 NO-TRACE-STRING
0 1 1 21 15 10 0 3 
.EQN 1 -30 528 0 0
{0:Xa8}NAME({0:n}NAME,{0:N}NAME)=
.EQN 14 0 529 0 0
{0:N}NAME:5
.EQN 0 7 530 0 0
{0:n}NAME:0;{0:N}NAME-1
.EQN 3 -7 531 0 0
{0:Xa8}NAME({0:n}NAME,{0:N}NAME)=
.EQN 5 29 669 0 0
&&(_n_u_l_l_&_n_u_l_l_)&|({0:Xa8}NAME({0:n}NAME,5)),0@&&(_n_u_l_l_&_n_u_l_l_)&{0:n}NAME
0 0 1 1 1 0 0 1 1 
0 0 1 1 1 0 0 1 1 
0 1 0 2 1 1 NO-TRACE-STRING
0 1 1 2 1 1 NO-TRACE-STRING
0 3 2 0 1 1 NO-TRACE-STRING
0 4 3 0 1 1 NO-TRACE-STRING
0 1 4 0 1 1 NO-TRACE-STRING
0 2 5 0 1 1 NO-TRACE-STRING
0 3 6 0 1 1 NO-TRACE-STRING
0 4 0 0 1 1 NO-TRACE-STRING
0 1 1 0 1 1 NO-TRACE-STRING
0 2 2 0 1 1 NO-TRACE-STRING
0 3 3 0 1 1 NO-TRACE-STRING
0 4 4 0 1 1 NO-TRACE-STRING
0 1 5 0 1 1 NO-TRACE-STRING
0 2 6 0 1 1 NO-TRACE-STRING
0 3 0 0 1 1 NO-TRACE-STRING
0 4 1 0 1 1 NO-TRACE-STRING
0 1 1 21 15 10 0 3 
.EQN 12 -29 532 0 0
{0:N}NAME:6
.EQN 0 7 533 0 0
{0:n}NAME:0;{0:N}NAME-1
.EQN 3 -7 534 0 0
{0:Xa8}NAME({0:n}NAME,{0:N}NAME)=
.EQN 8 29 672 0 0
&&(_n_u_l_l_&_n_u_l_l_)&|({0:Xa8}NAME({0:n}NAME,6)),0@&&(_n_u_l_l_&_n_u_l_l_)&{0:n}NAME
0 0 1 1 1 0 0 1 1 
0 0 1 1 1 0 0 1 1 
0 1 0 2 1 1 NO-TRACE-STRING
0 1 1 2 1 1 NO-TRACE-STRING
0 3 2 0 1 1 NO-TRACE-STRING
0 4 3 0 1 1 NO-TRACE-STRING
0 1 4 0 1 1 NO-TRACE-STRING
0 2 5 0 1 1 NO-TRACE-STRING
0 3 6 0 1 1 NO-TRACE-STRING
0 4 0 0 1 1 NO-TRACE-STRING
0 1 1 0 1 1 NO-TRACE-STRING
0 2 2 0 1 1 NO-TRACE-STRING
0 3 3 0 1 1 NO-TRACE-STRING
0 4 4 0 1 1 NO-TRACE-STRING
0 1 5 0 1 1 NO-TRACE-STRING
0 2 6 0 1 1 NO-TRACE-STRING
0 3 0 0 1 1 NO-TRACE-STRING
0 4 1 0 1 1 NO-TRACE-STRING
0 1 1 21 15 10 0 3 
.EQN 11 -29 535 0 0
{0:N}NAME:7
.EQN 0 7 536 0 0
{0:n}NAME:0;{0:N}NAME-1
.EQN 3 -7 537 0 0
{0:Xa8}NAME({0:n}NAME,{0:N}NAME)=
.EQN 9 29 676 0 0
&&(_n_u_l_l_&_n_u_l_l_)&|({0:Xa8}NAME({0:n}NAME,7)),0@&&(_n_u_l_l_&_n_u_l_l_)&{0:n}NAME
0 0 1 1 1 0 0 1 1 
0 0 1 1 1 0 0 1 1 
0 1 0 2 1 1 NO-TRACE-STRING
0 1 1 2 1 1 NO-TRACE-STRING
0 3 2 0 1 1 NO-TRACE-STRING
0 4 3 0 1 1 NO-TRACE-STRING
0 1 4 0 1 1 NO-TRACE-STRING
0 2 5 0 1 1 NO-TRACE-STRING
0 3 6 0 1 1 NO-TRACE-STRING
0 4 0 0 1 1 NO-TRACE-STRING
0 1 1 0 1 1 NO-TRACE-STRING
0 2 2 0 1 1 NO-TRACE-STRING
0 3 3 0 1 1 NO-TRACE-STRING
0 4 4 0 1 1 NO-TRACE-STRING
0 1 5 0 1 1 NO-TRACE-STRING
0 2 6 0 1 1 NO-TRACE-STRING
0 3 0 0 1 1 NO-TRACE-STRING
0 4 1 0 1 1 NO-TRACE-STRING
0 1 1 21 15 10 0 3 
.EQN 32 -29 538 0 0
{0:N}NAME:8
.EQN 0 7 539 0 0
{0:n}NAME:0;{0:N}NAME-1
.EQN 3 -7 540 0 0
{0:Xa8}NAME({0:n}NAME,{0:N}NAME)=
.EQN 0 32 678 0 0
&&(_n_u_l_l_&_n_u_l_l_)&|({0:Xa8}NAME({0:n}NAME,8)),0@&&(_n_u_l_l_&_n_u_l_l_)&{0:n}NAME
0 0 1 1 1 0 0 1 1 
0 0 1 1 1 0 0 1 1 
0 1 0 2 1 1 NO-TRACE-STRING
0 1 1 2 1 1 NO-TRACE-STRING
0 3 2 0 1 1 NO-TRACE-STRING
0 4 3 0 1 1 NO-TRACE-STRING
0 1 4 0 1 1 NO-TRACE-STRING
0 2 5 0 1 1 NO-TRACE-STRING
0 3 6 0 1 1 NO-TRACE-STRING
0 4 0 0 1 1 NO-TRACE-STRING
0 1 1 0 1 1 NO-TRACE-STRING
0 2 2 0 1 1 NO-TRACE-STRING
0 3 3 0 1 1 NO-TRACE-STRING
0 4 4 0 1 1 NO-TRACE-STRING
0 1 5 0 1 1 NO-TRACE-STRING
0 2 6 0 1 1 NO-TRACE-STRING
0 3 0 0 1 1 NO-TRACE-STRING
0 4 1 0 1 1 NO-TRACE-STRING
0 1 1 21 15 10 0 3 
.EQN 26 -32 541 0 0
{0:N}NAME:9
.EQN 0 7 542 0 0
{0:n}NAME:0;{0:N}NAME-1
.EQN 3 -7 543 0 0
{0:Xa8}NAME({0:n}NAME,{0:N}NAME)=
.EQN 1 32 681 0 0
&&(_n_u_l_l_&_n_u_l_l_)&|({0:Xa8}NAME({0:n}NAME,9)),0@&&(_n_u_l_l_&_n_u_l_l_)&{0:n}NAME
0 0 1 1 1 0 0 1 1 
0 0 1 1 1 0 0 1 1 
0 1 0 2 1 1 NO-TRACE-STRING
0 1 1 2 1 1 NO-TRACE-STRING
0 3 2 0 1 1 NO-TRACE-STRING
0 4 3 0 1 1 NO-TRACE-STRING
0 1 4 0 1 1 NO-TRACE-STRING
0 2 5 0 1 1 NO-TRACE-STRING
0 3 6 0 1 1 NO-TRACE-STRING
0 4 0 0 1 1 NO-TRACE-STRING
0 1 1 0 1 1 NO-TRACE-STRING
0 2 2 0 1 1 NO-TRACE-STRING
0 3 3 0 1 1 NO-TRACE-STRING
0 4 4 0 1 1 NO-TRACE-STRING
0 1 5 0 1 1 NO-TRACE-STRING
0 2 6 0 1 1 NO-TRACE-STRING
0 3 0 0 1 1 NO-TRACE-STRING
0 4 1 0 1 1 NO-TRACE-STRING
0 1 1 21 15 10 0 3 
.EQN 25 -32 544 0 0
{0:N}NAME:10
.EQN 0 7 545 0 0
{0:n}NAME:0;{0:N}NAME-1
.EQN 3 -7 546 0 0
{0:Xa8}NAME({0:n}NAME,{0:N}NAME)=
.EQN 1 31 685 0 0
&&(_n_u_l_l_&_n_u_l_l_)&|({0:Xa8}NAME({0:n}NAME,10)),0@&&(_n_u_l_l_&_n_u_l_l_)&{0:n}NAME
0 0 1 1 1 0 0 1 1 
0 0 1 1 1 0 0 1 1 
0 1 0 2 1 1 NO-TRACE-STRING
0 1 1 2 1 1 NO-TRACE-STRING
0 3 2 0 1 1 NO-TRACE-STRING
0 4 3 0 1 1 NO-TRACE-STRING
0 1 4 0 1 1 NO-TRACE-STRING
0 2 5 0 1 1 NO-TRACE-STRING
0 3 6 0 1 1 NO-TRACE-STRING
0 4 0 0 1 1 NO-TRACE-STRING
0 1 1 0 1 1 NO-TRACE-STRING
0 2 2 0 1 1 NO-TRACE-STRING
0 3 3 0 1 1 NO-TRACE-STRING
0 4 4 0 1 1 NO-TRACE-STRING
0 1 5 0 1 1 NO-TRACE-STRING
0 2 6 0 1 1 NO-TRACE-STRING
0 3 0 0 1 1 NO-TRACE-STRING
0 4 1 0 1 1 NO-TRACE-STRING
0 1 1 21 15 10 0 3 
.EQN 41 -31 547 0 0
{0:N}NAME:11
.EQN 0 7 548 0 0
{0:n}NAME:0;{0:N}NAME-1
.EQN 3 -7 549 0 0
{0:Xa8}NAME({0:n}NAME,{0:N}NAME)=
.EQN 0 17 596 0 0
|({0:Xa8}NAME({0:n}NAME,{0:N}NAME))=
.EQN 4 16 687 0 0
&&(_n_u_l_l_&_n_u_l_l_)&|({0:Xa8}NAME({0:n}NAME,11)),0@&&(_n_u_l_l_&_n_u_l_l_)&{0:n}NAME
0 0 1 1 1 0 0 1 1 
0 0 1 1 1 0 0 1 1 
0 1 0 2 1 1 NO-TRACE-STRING
0 1 1 2 1 1 NO-TRACE-STRING
0 3 2 0 1 1 NO-TRACE-STRING
0 4 3 0 1 1 NO-TRACE-STRING
0 1 4 0 1 1 NO-TRACE-STRING
0 2 5 0 1 1 NO-TRACE-STRING
0 3 6 0 1 1 NO-TRACE-STRING
0 4 0 0 1 1 NO-TRACE-STRING
0 1 1 0 1 1 NO-TRACE-STRING
0 2 2 0 1 1 NO-TRACE-STRING
0 3 3 0 1 1 NO-TRACE-STRING
0 4 4 0 1 1 NO-TRACE-STRING
0 1 5 0 1 1 NO-TRACE-STRING
0 2 6 0 1 1 NO-TRACE-STRING
0 3 0 0 1 1 NO-TRACE-STRING
0 4 1 0 1 1 NO-TRACE-STRING
0 1 1 21 15 10 0 3 
.EQN 27 -33 550 0 0
{0:N}NAME:12
.EQN 0 7 551 0 0
{0:n}NAME:0;{0:N}NAME-1
.EQN 3 -7 552 0 0
{0:Xa8}NAME({0:n}NAME,{0:N}NAME)=
.EQN 0 13 593 0 0
|({0:Xa8}NAME({0:n}NAME,{0:N}NAME))=
.EQN 5 20 689 0 0
&&(_n_u_l_l_&_n_u_l_l_)&|({0:Xa8}NAME({0:n}NAME,12)),0@&&(_n_u_l_l_&_n_u_l_l_)&{0:n}NAME
0 0 1 1 1 0 0 1 1 
0 0 1 1 1 0 0 1 1 
0 1 0 2 1 1 NO-TRACE-STRING
0 1 1 2 1 1 NO-TRACE-STRING
0 3 2 0 1 1 NO-TRACE-STRING
0 4 3 0 1 1 NO-TRACE-STRING
0 1 4 0 1 1 NO-TRACE-STRING
0 2 5 0 1 1 NO-TRACE-STRING
0 3 6 0 1 1 NO-TRACE-STRING
0 4 0 0 1 1 NO-TRACE-STRING
0 1 1 0 1 1 NO-TRACE-STRING
0 2 2 0 1 1 NO-TRACE-STRING
0 3 3 0 1 1 NO-TRACE-STRING
0 4 4 0 1 1 NO-TRACE-STRING
0 1 5 0 1 1 NO-TRACE-STRING
0 2 6 0 1 1 NO-TRACE-STRING
0 3 0 0 1 1 NO-TRACE-STRING
0 4 1 0 1 1 NO-TRACE-STRING
0 1 1 21 15 10 0 3 
.EQN 60 -33 571 0 0
{0:N}NAME:13
.EQN 0 7 572 0 0
{0:n}NAME:0;{0:N}NAME-1
.EQN 3 -7 573 0 0
{0:Xa8}NAME({0:n}NAME,{0:N}NAME)=
.EQN 0 19 590 0 0
|({0:Xa8}NAME({0:n}NAME,{0:N}NAME))=
.EQN 5 15 691 0 0
&&(_n_u_l_l_&_n_u_l_l_)&|({0:Xa8}NAME({0:n}NAME,13)),0@&&(_n_u_l_l_&_n_u_l_l_)&{0:n}NAME
0 0 1 1 1 0 0 1 1 
0 0 1 1 1 0 0 1 1 
0 1 0 2 1 1 NO-TRACE-STRING
0 1 1 2 1 1 NO-TRACE-STRING
0 3 2 0 1 1 NO-TRACE-STRING
0 4 3 0 1 1 NO-TRACE-STRING
0 1 4 0 1 1 NO-TRACE-STRING
0 2 5 0 1 1 NO-TRACE-STRING
0 3 6 0 1 1 NO-TRACE-STRING
0 4 0 0 1 1 NO-TRACE-STRING
0 1 1 0 1 1 NO-TRACE-STRING
0 2 2 0 1 1 NO-TRACE-STRING
0 3 3 0 1 1 NO-TRACE-STRING
0 4 4 0 1 1 NO-TRACE-STRING
0 1 5 0 1 1 NO-TRACE-STRING
0 2 6 0 1 1 NO-TRACE-STRING
0 3 0 0 1 1 NO-TRACE-STRING
0 4 1 0 1 1 NO-TRACE-STRING
0 1 1 21 15 10 0 3 
.EQN 29 -34 559 0 0
{0:N}NAME:14
.EQN 0 7 560 0 0
{0:n}NAME:0;{0:N}NAME-1
.EQN 3 -7 561 0 0
{0:Xa8}NAME({0:n}NAME,{0:N}NAME)=
.EQN 0 18 588 0 0
|({0:Xa8}NAME({0:n}NAME,{0:N}NAME))=
.EQN 4 13 693 0 0
&&(_n_u_l_l_&_n_u_l_l_)&|({0:Xa8}NAME({0:n}NAME,14)),0@&&(_n_u_l_l_&_n_u_l_l_)&{0:n}NAME
0 0 1 1 1 0 0 1 1 
0 0 1 1 1 0 0 1 1 
0 1 0 2 1 1 NO-TRACE-STRING
0 1 1 2 1 1 NO-TRACE-STRING
0 3 2 0 1 1 NO-TRACE-STRING
0 4 3 0 1 1 NO-TRACE-STRING
0 1 4 0 1 1 NO-TRACE-STRING
0 2 5 0 1 1 NO-TRACE-STRING
0 3 6 0 1 1 NO-TRACE-STRING
0 4 0 0 1 1 NO-TRACE-STRING
0 1 1 0 1 1 NO-TRACE-STRING
0 2 2 0 1 1 NO-TRACE-STRING
0 3 3 0 1 1 NO-TRACE-STRING
0 4 4 0 1 1 NO-TRACE-STRING
0 1 5 0 1 1 NO-TRACE-STRING
0 2 6 0 1 1 NO-TRACE-STRING
0 3 0 0 1 1 NO-TRACE-STRING
0 4 1 0 1 1 NO-TRACE-STRING
0 1 1 21 15 10 0 3 
.EQN 60 -30 574 0 0
{0:N}NAME:15
.EQN 0 7 575 0 0
{0:n}NAME:0;{0:N}NAME-1
.EQN 3 -7 582 0 0
{0:Xa8}NAME({0:n}NAME,{0:N}NAME)=
.EQN 0 20 585 0 0
|({0:Xa8}NAME({0:n}NAME,{0:N}NAME))=
.EQN 5 11 695 0 0
&&(_n_u_l_l_&_n_u_l_l_)&|({0:Xa8}NAME({0:n}NAME,15)),0@&&(_n_u_l_l_&_n_u_l_l_)&{0:n}NAME
0 0 1 1 1 0 0 1 1 
0 0 1 1 1 0 0 1 1 
0 1 0 2 1 1 NO-TRACE-STRING
0 1 1 2 1 1 NO-TRACE-STRING
0 3 2 0 1 1 NO-TRACE-STRING
0 4 3 0 1 1 NO-TRACE-STRING
0 1 4 0 1 1 NO-TRACE-STRING
0 2 5 0 1 1 NO-TRACE-STRING
0 3 6 0 1 1 NO-TRACE-STRING
0 4 0 0 1 1 NO-TRACE-STRING
0 1 1 0 1 1 NO-TRACE-STRING
0 2 2 0 1 1 NO-TRACE-STRING
0 3 3 0 1 1 NO-TRACE-STRING
0 4 4 0 1 1 NO-TRACE-STRING
0 1 5 0 1 1 NO-TRACE-STRING
0 2 6 0 1 1 NO-TRACE-STRING
0 3 0 0 1 1 NO-TRACE-STRING
0 4 1 0 1 1 NO-TRACE-STRING
0 1 1 21 15 10 0 3 
.EQN 36 -33 578 0 0
{0:N}NAME:16
.EQN 0 7 579 0 0
{0:n}NAME:0;{0:N}NAME-1
.EQN 3 -7 580 0 0
{0:Xa8}NAME({0:n}NAME,{0:N}NAME)=
.EQN 10 33 697 0 0
&&(_n_u_l_l_&_n_u_l_l_)&|({0:Xa8}NAME({0:n}NAME,16)),0@&&(_n_u_l_l_&_n_u_l_l_)&{0:n}NAME
0 0 1 1 1 0 0 1 1 
0 0 1 1 1 0 0 1 1 
0 1 0 2 1 1 NO-TRACE-STRING
0 1 1 2 1 1 NO-TRACE-STRING
0 3 2 0 1 1 NO-TRACE-STRING
0 4 3 0 1 1 NO-TRACE-STRING
0 1 4 0 1 1 NO-TRACE-STRING
0 2 5 0 1 1 NO-TRACE-STRING
0 3 6 0 1 1 NO-TRACE-STRING
0 4 0 0 1 1 NO-TRACE-STRING
0 1 1 0 1 1 NO-TRACE-STRING
0 2 2 0 1 1 NO-TRACE-STRING
0 3 3 0 1 1 NO-TRACE-STRING
0 4 4 0 1 1 NO-TRACE-STRING
0 1 5 0 1 1 NO-TRACE-STRING
0 2 6 0 1 1 NO-TRACE-STRING
0 3 0 0 1 1 NO-TRACE-STRING
0 4 1 0 1 1 NO-TRACE-STRING
0 1 1 21 15 10 0 3 
.TXT 47 -33 584 0 0
Cg a71.000000,71.000000,359
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue255;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard For N = 8, 16, the 
nonzero values occur at n = N/8 and n = 7N/8.  The corresponding \par 
analog frequencies are (fs)n/N.  For example, for N = 16, (fs)(N/8)/N 
= (8)(16/8)/16 = 1Hz,\par (fs)(7N/8)/N = (8)(7*16/8)/16 = 7Hz.  The 
values at n = N/8 and n = 7N/8 can be normalized\par by dividing by N.  
For example, for N = 16, 16/16 = 1 at n = 16/8 = 2 and n = 7*16/8 = 
14. }
.TXT 11 0 586 0 0
Cg a72.000000,72.000000,367
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue255;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard In general, the for 
number of samples N which is the multiple of the sampling rate fs, the
\par n values where the nonzero spectrum occur are n = N/fs and n = 
(fs-1)*N/fs.  The \par corresponding analog frequencies are fs*n/N.  
The values at n = N/fs and n = (fs-1)*N/fs \par can be normalized by 
dividing by N.  The value after the normalization is equal to the\par 
analog spectrum. }
.TXT 12 0 597 0 0
Cg a72.000000,72.000000,548
{\rtf\ansi \deff0{\colortbl;\red0\green0\blue255;}{\fonttbl{\f0
\fcharset0\fnil Arial;}}\plain\cf1\fs20 \pard When the number of 
samples N is not the multiple of the sampling rate fs, the DFT of a
\par single cosine does not yield two impulses as desired.  Instead, 
the spectrums show \par peaks around the n values close to frequencies 
fs*n/N = 1 (analog frequencies).  When N\par value is close to the 
multiple of the sampling rate, the peaks approach impulse.  The\par 
spectrum leakage is larger when the number of samples N is not close 
to the multiple\par of the sampling rate.  When N is large, the 
spectrum leakage is small, and the peaks\par approach impulses as 
shown below.}
.EQN 18 0 605 0 0
{0:Ts}NAME:0.125
.EQN 0 9 606 0 0
{0:k}NAME:0;50
.EQN 0 9 607 0 0
{0:x3}NAME({0:k}NAME):2*{0:cos}NAME(2*{0:\p}NAME*{0:k}NAME*{0:Ts}NAME)
.EQN 6 -18 608 0 0
{0:Xa8}NAME({0:n}NAME,{0:N}NAME):((0,{0:N}NAME-1,{0:k}NAME,{0:x3}NAME({0:k}NAME)*({0:e}NAME)^(-1j*{0:n}NAME*(2*{0:\p}NAME)/({0:N}NAME)*{0:k}NAME)){64})
.EQN 6 0 609 0 0
{0:N}NAME:50
.EQN 0 7 610 0 0
{0:n}NAME:0;{0:N}NAME-1
.EQN 3 -7 611 0 0
{0:Xa8}NAME({0:n}NAME,{0:N}NAME)=
.EQN 4 18 614 0 0
&&(_n_u_l_l_&_n_u_l_l_)&|({0:Xa8}NAME({0:n}NAME,{0:N}NAME))@&&(_n_u_l_l_&_n_u_l_l_)&{0:n}NAME
0 0 1 1 1 0 0 1 1 
0 0 1 1 1 0 0 1 1 
0 1 0 0 1 1 NO-TRACE-STRING
0 2 1 0 1 1 NO-TRACE-STRING
0 3 2 0 1 1 NO-TRACE-STRING
0 4 3 0 1 1 NO-TRACE-STRING
0 1 4 0 1 1 NO-TRACE-STRING
0 2 5 0 1 1 NO-TRACE-STRING
0 3 6 0 1 1 NO-TRACE-STRING
0 4 0 0 1 1 NO-TRACE-STRING
0 1 1 0 1 1 NO-TRACE-STRING
0 2 2 0 1 1 NO-TRACE-STRING
0 3 3 0 1 1 NO-TRACE-STRING
0 4 4 0 1 1 NO-TRACE-STRING
0 1 5 0 1 1 NO-TRACE-STRING
0 2 6 0 1 1 NO-TRACE-STRING
0 3 0 0 1 1 NO-TRACE-STRING
0 4 1 0 1 1 NO-TRACE-STRING
0 1 1 36 15 10 0 3