I am wondering whether I can get the upper bound in closed form of

$\sum_{n=1}^N(\alpha \exp(j2\pi n/N))$ where $\alpha = +1\ or -1$ and $j^2=-1$.

If alpha is just positive one, this would be just a single value,

but I'm trying to get the upper bound when alpha is +1 or -1, randomly,

while the total amount of +1 and -1 is different at most 1. (N/2 or N/2+1)

I have looked for exponential sums materials, but can't see things like this.

edit : $\alpha $ is generated by LFSR, so it holds pseudo randomness property.
or equivalently, it is a maximal length sequences.

I am wondering whether I can get the upper bound in closed form of

$$\sum_{n=1}^N(\alpha \exp(j2\pi n/N)) \text{ where } \alpha = +1\text{ or }-1 \text{ and } j^2=-1.$$

If alpha is just positive one, this would be just a single value,

but I'm trying to get the upper bound when alpha is $+1$ or $-1$, randomly,

while the total amount of $+1$ and $-1$ is different at most $1.$ ($N/2$ or $N/2+1$)

I have looked for exponential sums materials, but can't see things like this.

edit : $\alpha $ is generated by LFSR, so it holds pseudo randomness property.
or equivalently, it is a maximal length sequences.

I am wondering whether I can get the upper bound in closed form,

$\sum_n(\alpha \exp(j2\pi n/N))$ where $\alpha = +1\ or -1$

If alpha is just positive one, this would be just a single value,

but I'm trying to get the upper bound when alpha is +1 or -1, randomly,

while the total amount of +1 and -1 is different at most 1. (N/2 or N/2+1)

I have looked for exponential sums materials, but can't see things like this.

edit : $\alpha $ is generated by LFSR, so it holds pseudo randomness property.
or equivalently, it is a maximal length sequences.

I am wondering whether I can get the upper bound in closed form of

$\sum_{n=1}^N(\alpha \exp(j2\pi n/N))$ where $\alpha = +1\ or -1$ and $j^2=-1$.

If alpha is just positive one, this would be just a single value,

but I'm trying to get the upper bound when alpha is +1 or -1, randomly,

while the total amount of +1 and -1 is different at most 1. (N/2 or N/2+1)

I have looked for exponential sums materials, but can't see things like this.

edit : $\alpha $ is generated by LFSR, so it holds pseudo randomness property.
or equivalently, it is a maximal length sequences.