# Why two peaks in fft of sine wave

Started by July 12, 2003
```Hi

I had made a fft c program and it is working perfectly for discrete
values. I have also checked it on MATLAB and quite sure that the
program is correct.

However I have two doubts.

I inputted the samples of a sine wave of 1000 Hz with a sampling
frequency 4000Hz. For a simple 16 point FFT I got two peaks one at
x[4] and other at x[12]. Now shouldn't a simple sine wave show just
one peak at 1000Hz.

Is it that the values I get are centered along the imaginary axis. I
am not able to understand where the frequency samples after doing fft
actually lie in the spectrum.

Thanks
Aarul Jain
```
```"Aarul Jain" <aaruljain@yahoo.com> wrote in message
> Hi
>
> I had made a fft c program and it is working perfectly for discrete
> values. I have also checked it on MATLAB and quite sure that the
> program is correct.
>
> However I have two doubts.
>
> I inputted the samples of a sine wave of 1000 Hz with a sampling
> frequency 4000Hz. For a simple 16 point FFT I got two peaks one at
> x[4] and other at x[12]. Now shouldn't a simple sine wave show just
> one peak at 1000Hz.
>
> Is it that the values I get are centered along the imaginary axis. I
> am not able to understand where the frequency samples after doing fft
> actually lie in the spectrum.
>

Congratulations, it appears to be working properly!
The FFT generates coefficients for complex exponentials.
To get a sine, you need the sum of two complex exponentials - one at
positive frequency and one at negative frequency.
The FFT generates a single replica of a periodic spectrum.  So you can think
of the result as being plotted on a circular axis that repeats at
x[0]=x[16].
The positive frequency component shows up 4 above x[0] or x[4]
The negative frequency shows up 4 below x[16] or x[12].

Since the time function is real, the even part will transform to an even
real function in frequency and the odd part will transform to an odd
imaginary part in frequency.
So, if the input was really sin(k*t) then you should have an odd, imaginary
output.
If the value at x[4] is j0.5 then the value at x[12] would be -j0.5.  So,
you really only need to know the N/2 values from x[0] to x[7] to know the
whole thing.  But, most FFTs allow complex inputs and provide the full N
outputs.

Fred

```
```Thanks for your good help.

Regards

Aarul Jain

"Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote in message
news:<2IYPa.2572\$Jk5.1759312@feed2.centurytel.net>...
> "Aarul Jain" <aaruljain@yahoo.com> wrote in message
> > Hi
> >
> > I had made a fft c program and it is working perfectly for discrete
> > values. I have also checked it on MATLAB and quite sure that the
> > program is correct.
> >
> > However I have two doubts.
> >
> > I inputted the samples of a sine wave of 1000 Hz with a sampling
> > frequency 4000Hz. For a simple 16 point FFT I got two peaks one at
> > x[4] and other at x[12]. Now shouldn't a simple sine wave show just
> > one peak at 1000Hz.
> >
> > Is it that the values I get are centered along the imaginary axis. I
> > am not able to understand where the frequency samples after doing fft
> > actually lie in the spectrum.
> >
>
> Congratulations, it appears to be working properly!
> The FFT generates coefficients for complex exponentials.
> To get a sine, you need the sum of two complex exponentials - one at
> positive frequency and one at negative frequency.
> The FFT generates a single replica of a periodic spectrum.  So you can think
> of the result as being plotted on a circular axis that repeats at
> x[0]=x[16].
> The positive frequency component shows up 4 above x[0] or x[4]
> The negative frequency shows up 4 below x[16] or x[12].
>
> Since the time function is real, the even part will transform to an even
> real function in frequency and the odd part will transform to an odd
> imaginary part in frequency.
> So, if the input was really sin(k*t) then you should have an odd, imaginary
> output.
> If the value at x[4] is j0.5 then the value at x[12] would be -j0.5.  So,
> you really only need to know the N/2 values from x[0] to x[7] to know the
> whole thing.  But, most FFTs allow complex inputs and provide the full N
> outputs.
>
> Fred
```