Discrete Fourier Transform of Sine Function

[DuncanM]

(1) For a real function, g(x), the Fourier integral transform is defined by
$$g(x) = \int_{0}^{\infty} A(\omega )cos(2\pi \omega x)d\omega - \int_{0}^{\infty} B(\omega )sin(2\pi \omega x)d\omega$$

where
$$A(\omega ) = 2 \int_{-\infty}^{\infty} g(x)cos(2\pi \omega x)dx$$

and
$$B(\omega ) = 2 \int_{-\infty}^{\infty} g(x)sin(2\pi \omega x)dx$$

For an even function, B(ω) = 0
For an odd function A(ω) = 0

(2) For discrete points, the function is declared as
$$g_{j} = a_{0} + \sum_{k = 1}^{[N/2]} a_{k} cos(kj\frac{2\pi}{N}) + \sum_{k = 1}^{[N/2]} b_{k} sin(kj\frac{2\pi}{N})$$

where
$$a_{k} = \frac{2}{N} \sum_{j = 0}^{N-1} g_{j} cos(jk\frac{2\pi}{N})$$

and
$$b_{k} = \frac{2}{N} \sum_{j = 0}^{N-1} g_{j} sin(jk\frac{2\pi}{N})$$

The text I am reading states that the discrete coefficients approximate the exact Fourier coefficients of a periodic function.

Now, taking a baby step, say I consider the function g(x) = sin(x), the plain, old sine function.
Since the sine function is odd, the Fourier integral transform indicates that A(ω) = 0.
Since the exact Fourier coefficients of a periodic function approximate discrete coefficients, I expected the $a_{k}$ values of a DFT to be 0. But they are not. In fact, substituting the results of a DFT back into the original equation do not work without both the $a_{k}$ and $b_{k}$ values.

Here is what I did:
I assumed a period of 1 time unit, over 1 period, and divided the motion into uniform intervals. Calculating the sine values at each interval, this data was then fed into a DFT. The resultant transform includes both sine and cosine coefficients.

Shouldn't it include only coefficients for the sine terms?
What is going on? Have I made an incorrect leap in logic?

 

[jbunniii]

So you are setting $g_j = \sin(j)$? This does not result in an odd function when you extend it periodically. To see this, observe that $g_1 = \sin(1)$ whereas $g_{-1} = g_{N-1} = \sin(N-1)$, which does not equal $-\sin(1)$. Therefore, $g_1 \neq -g_{-1}$.

Instead, try setting $g_j = \sin(2\pi j / N)$, or more generally, $g_j = \sin(2\pi j k /N)$ where $k$ is any integer.

 

[DuncanM]

Here is some sample data. I wanted to have the points at t = T/4 and t = 3T/4 included, so the motion was divided into 20 uniform intervals.
T = 1 time unit.

k =  0    t = 0.0       sin(2*PI/20*t) =  0.0
k =  1    t =  0.05     sin(2*PI/20*t) =  0.309016994374947
k =  2    t =  0.1      sin(2*PI/20*t) =  0.587785252292473
k =  3    t =  0.15     sin(2*PI/20*t) =  0.809016994374947
k =  4    t =  0.2      sin(2*PI/20*t) =  0.951056516295154
k =  5    t =  0.25     sin(2*PI/20*t) =  1
k =  6    t =  0.3      sin(2*PI/20*t) =  0.951056516295154
k =  7    t =  0.35     sin(2*PI/20*t) =  0.809016994374947
k =  8    t =  0.4      sin(2*PI/20*t) =  0.587785252292473
k =  9    t =  0.45     sin(2*PI/20*t) =  0.309016994374948
k =  10   t =  0.5      sin(2*PI/20*t) =  0
k =  11   t =  0.55     sin(2*PI/20*t) =  -0.309016994374948
k =  12   t =  0.6      sin(2*PI/20*t) =  -0.587785252292473
k =  13   t =  0.65     sin(2*PI/20*t) =  -0.809016994374947
k =  14   t =  0.7      sin(2*PI/20*t) =  -0.951056516295154
k =  15   t =  0.75     sin(2*PI/20*t) =  -1
k =  16   t =  0.8      sin(2*PI/20*t) =  -0.951056516295154
k =  17   t =  0.85     sin(2*PI/20*t) =  -0.809016994374947
k =  18   t =  0.9      sin(2*PI/20*t) =  -0.587785252292473
k =  19   t =  0.95     sin(2*PI/20*t) =  -0.309016994374948
k =  20   t = 1.0       sin(2*PI/20*t) =   0

Taking just the sine values:

0.0
0.309016994374947
0.587785252292473
0.809016994374947
0.951056516295154
1
0.951056516295154
0.809016994374947
0.587785252292473
0.309016994374948
0
-0.309016994374948
-0.587785252292473
-0.809016994374947
-0.951056516295154
-1
-0.951056516295154
-0.809016994374947
-0.587785252292473
-0.309016994374948
0

and inputting them into a DFT, the following results are returned:

The a_k values follow:

a₀ = 0
0.14475387005697557
-0.020485172200211867
-0.016913755515248936
-0.015944715871194427
-0.015536943989308476
-0.015328780935943866
-0.015211428505699521
-0.015142744411310508
-0.015103970855542756
-0.015086357772515164

The b_k values follow:

0.9603791769948659
-0.06641129715700931
-0.03512177522380782
-0.02338660289058022
-0.01674484324556045
-0.01222429488042325
-0.008782322342524462
-0.005943089235252812
-0.003447382785156719
-0.0011305662206714806

Shouldn't the a_k values be 0? I mean, the DFT was performed on a simple sine function, and it doesn't get any more "pure" sine than that. Yet the DFT seems to indicate it includes some cosine factors. Is that correct, or am I making some wrong interpretation?