question on taking FFT of a negative-time signal?

Hi all,

I want to ask about taking FFT of a signal discrete-time whose indices are 
limited in [-6, +6] and with 13 data points.

x[n]=
1, for n=-2 .. +2;
0, for other n in [-6, +6]

However, in Matlab all indices start from 1, so I have to input this 
sequence as

x=[0 0 0 0 1 1 1 1 1 0 0 0 0];

y=fftshift(fft(x));

since the indeces have been shifted, so the y has no zero imaginary part(the 
original input sequence is real and even so y should have zero imaginary 
parts)

Now how to show the DFT of the original double-sided signal from -6 to 6? 
How to make y's imaginary parts zero?

-----------------------------------------

I have done the following experiments by using the shifting theorem: but I 
just could not make y's imaginary part zero... what's the probem?

>> y1=y.*exp(1j*2*pi*6*[0:length(y)-1]/13);
>> imag(y1)

ans =

  Columns 1 through 6

   -0.8230    0.2541    1.1974    0.6957   -1.4165   -3.8786

  Columns 7 through 12

   -4.9635   -3.8786   -1.4165    0.6957    1.1974    0.2541

  Column 13

   -0.8230

>> y1=y.*exp(-1j*2*pi*6*[0:length(y)-1]/13);
>> imag(y1)

ans =

  Columns 1 through 6

   -0.8230    0.2106    0.5606   -0.0000    0.6631    3.2154

  Columns 7 through 12

    4.9635    3.6531    0.9462   -0.1677    0.2887    0.1697

  Column 13

   -0.7752

>> y1=y.*exp(-1j*2*pi*6*[1:length(y)]/13);
>> imag(y1)

ans =

  Columns 1 through 6

    0.7752   -0.1697   -0.2887    0.1677   -0.9462   -3.6531

  Columns 7 through 12

   -4.9635   -3.2154   -0.6631   -0.0000   -0.5606   -0.2106

  Column 13

    0.8230

>> y1=y.*exp(1j*2*pi*6*[1:length(y)]/13);
>> imag(y1)

ans =

  Columns 1 through 6

    0.8230   -0.2541   -1.1974   -0.6957    1.4165    3.8786

  Columns 7 through 12

    4.9635    3.8786    1.4165   -0.6957   -1.1974   -0.2541

  Column 13

    0.8230

>>

"kiki" <lunaliu3@yahoo.com> wrote in message
news:co9dg5$hs3$1@news.Stanford.EDU...
> Hi all,
>
> I want to ask about taking FFT of a signal discrete-time whose indices are
> limited in [-6, +6] and with 13 data points.
>
> x[n]=
> 1, for n=-2 .. +2;
> 0, for other n in [-6, +6]
>
> However, in Matlab all indices start from 1, so I have to input this
> sequence as
>
> x=[0 0 0 0 1 1 1 1 1 0 0 0 0];
>
> y=fftshift(fft(x));
>
> since the indeces have been shifted, so the y has no zero imaginary
part(the
> original input sequence is real and even so y should have zero imaginary
> parts)
>
> Now how to show the DFT of the original double-sided signal from -6 to 6?
> How to make y's imaginary parts zero?
>
> -----------------------------------------
>

Hello Kiki,

Look up the Heaviside shifting theorem for Fourier transforms and apply it
to your fft's output.

Clay

kiki wrote:
> Hi all,
> 
> I want to ask about taking FFT of a signal discrete-time whose indices are 
> limited in [-6, +6] and with 13 data points.
> 
> x[n]=
> 1, for n=-2 .. +2;
> 0, for other n in [-6, +6]
> 
> However, in Matlab all indices start from 1, so I have to input this 
> sequence as
> 
> x=[0 0 0 0 1 1 1 1 1 0 0 0 0];
> 
> y=fftshift(fft(x));

You are mistaken in your mapping.  Matlab's x(1) corresponds to the time 
sequence's x_0, x(2) <-> x_1, etc.  For negative indecies you have to 
take it mod(N), so x(13) corresponds to x_{-1}.  Your vector should be

x = [1 1 1 0 0 0 0 0 0 0 0 1 1];
> 
> since the indeces have been shifted, so the y has no zero imaginary part(the 
> original input sequence is real and even so y should have zero imaginary 
> parts)
> 
> Now how to show the DFT of the original double-sided signal from -6 to 6? 
> How to make y's imaginary parts zero?
> 
> -----------------------------------------
> 
> I have done the following experiments by using the shifting theorem: but I 
> just could not make y's imaginary part zero... what's the probem?
>  
You are shifting by the wrong amount.  You shifted the original sample 
by 1/2 the overall cycle time, try this:

fft(x) .* exp(0.5 * -1j*2*pi*[0:12]/13)

If you don't have it already this question could be answered in much 
more depth by reading Oppenheim, Willsky & Young, "Signals and Systems" 
or other similar books.

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com