# 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.

• I have a few queries now about the pulse shaping: Based on what I understood, after OQPSK Modultaion, I should apply a pulse shaping filter on the sequence before upsampling it. While pulse shaping, I should specify the no. of samples per symbol, filter order and roll off factor. Should the no. of samples be (sampling rate/symbol rate)? I assume that it is safe to use a roll off factor of 0.5 and how should I decide the filter order? May 21 '15 at 8:45