Suppose that the received signal in a direct-sequence spread-spectrum
communications system is $r(t) = \pm As(t) + \mathcal N(t)$ where $s(t) = \sum_{n=0}^{N-1} x_n p(t-nT_c)$ is the PN signal of duration $T = NT_c$ that is used to spread the spectrum
(the $x_n$ have value $\pm 1$, $N$ is the number of "chips" in the
sequence, $p(t)$ is the unit-power baseband chip pulse shape of duration
$T_c$) and $\mathcal N(t)$ is
white Gaussian noise with two-sided power spectral density $\frac{N_0}{2}$.
As @JasonR's cogent comment points out, this is just a canonical
"antipodal signaling in white Gaussian noise" system using
antipodal signals $\pm As(t)$ and we can ignore the fact that $s(t)$
is not a rectangular pulse or a raised cosine pulse or what-have-you:
all the antipodal signaling analysis and methodology holds. The
receiver doesn't
just multiply $r(t)$ by $s(t)$, it multiplies and then integrates the product $r(t)\cdot s(t)$. That is, we correlate $r(t)$ with the
locally-generated spreading signal $s(t)$:
\begin{align}
Y &= \int_{0}^T r(t)\cdot s(t)\ \mathrm dt \tag{1}\\
&= \int_{0}^T \pm As(t)\cdot s(t)\ \mathrm dt
+ \int_{0}^T \mathcal N(t)\cdot s(t)\ \mathrm dt\\
&= \pm AT + \eta \tag{2}
\end{align}
where $\eta$ is a zero-mean Gaussian random variable with
variance $\frac{N_0T}{2}$. The bit error probability is thus
$$P_e = Q\left(\frac{AT}{\sqrt{\frac{N_0T}{2}}}\right)
= Q\left(\sqrt{\frac{2A^2T}{N_0}}\right)
= Q\left(\sqrt{\frac{2\mathscr E_b}{N_0}}\right)$$
as with the usual antipodal signaling.
With direct-sequence spread-spectrum signaling, an alternative
implementation of the receiver uses a chip correlator followed
by digital correlation. The receiver now computes
\begin{align}
Y_n &= \int_{nT_c}^{(n+1)T_c} r(t)\cdot p(t-nT)\ \mathrm dt,
\quad n = 0, 1, \ldots N-1 \tag{3}\\
&= \int_{nT_c}^{(n+1)T_c} \pm As(t)\cdot p(t-nT)\ \mathrm dt
+ \int_{nT_c}^{(n+1)T_c} \mathcal N(t)\cdot p(t-nT)\ \mathrm dt\\
&= \pm Ax_nT_c + \eta_n \tag{4}
\end{align}
where the $\eta_n$ are zero-mean Gaussian random variables.
The receiver now does a digital correlation by computing
$$\sum_{n=0}^{N-1} Y_n \cdot x_n = \sum_{n=0}^{N-1}
\pm Ax_nT_c\cdot x_n + \eta_n\cdot x_n = Y + \eta \tag{5}$$
and makes a decision on $Y$ just as before.
It is a
major mistake to make individual decisions $\hat{x}_n$ on the chip
variables $Y_n$ and then sum
$$\sum_{n=0}^{N-1} \hat{x}_n \cdot x_n.$$
The sum is easy to compute because the $\hat{x}n$ and the $x_n$
have value $\pm 1$ but the error probability of those decisions is
$$P_{e, chip} = Q\left(\frac{AT_c}{\sqrt{\frac{N_0T_c}{2}}}\right)
= Q\left(\sqrt{\frac{2A^2T_c}{N_0}}\right)
= Q\left(\sqrt{\frac{2\mathscr E_c}{N_0}}\right)$$
where the chip energy $\mathscr E_c$ is smaller than
$\mathscr E_b$ by a factor of $N$. Even when the bit SNR
is reasonably large, the chip SNR is very small and the
chip decisions are very little different from random guesses.
Consequently, hard chip decisions are the wrong way to go, it
is far far better to keep the soft decisions $Y_n$ and use them
in a digital correlation to create the decision variable that is
the optimum one to use.
