Frequency Spectrum of a sequence The OQPSK modulator in Matlab produces an output sequence as follows: 
[0.707
0.707+0.707i
-0.707-0.707i
...and so on
] 
This is the sequence and I am trying to view the frequency spectrum of this sequence. With the sampling rate being 2 MHz, I expect to see a peak centered at 0 with a bandwidth of 2 MHz. When I upsample the above sequence and multiply this sequence with a carrier of say 2.4 GHz, I should have 2 peaks centered at -2.4 GHz and 2.4 GHz and zeros at other frequencies. But somehow, the spectrum is distributed between -2.4 and 2.4 GHz. Am I making a mistake in plotting the frequency spectrum or is there something wrong in what I understood?
You're missing a step: pulse shaping. The sequence [.7+.7i, ...] is a sequence of pulse amplitudes. As such, it should not be considered to be signal; in consequence, it does not have a spectrum.

Say your sequence of amplitudes is $A=a_0, a_1, a_2,\ldots$. Now, you need to specify your symbol rate (how many pulses per second you'll transmit). Call this rate $R_p$ and let $T_p=1/R_p$ be the pulse interval. Finally, you need to specify a pulse $p(t)$ that has this property: $$\int_{-\infty}^\infty p(t-kT_p)p(t-lT_p)dt=\begin{cases}1,\textrm{ if $k=l$}\\0,\textrm{ if $k\neq l$}\end{cases}$$for $k$ and $l$ integers. Now you're ready to create your actual signal: $$s(t)=\sum_i a_ip(t-iT_p).$$ This signal:


Has a spectrum; specifically, its bandwidth is the same as the bandwidth of the pulse $p(t)$.
Can be up-converted to a carrier frequency.
Can be converted to an analog signal and physically transmitted over a medium.


For completeness, I'll explain how to recover the amplitudes $a_i$ from $s(t)$ (ignoring noise for simplicity). The receiver can estimate $a_k$ by doing this operation: $$\hat{a_k}=\int_{-\infty}^\infty s(t)p(t-kTp).$$ You can easily verifty that this is the case, using the properties of $p(t)$. Such an operation can be implemented with a matched filter.