How much noise gets through my raised cosine filter?
For my purpose, the correct method should normalize the filter so that the sum of the filter coefficients is 1. The DC value is normalized because this is a low-pass filter. It would be different if the filter were something else, like pass-band.

Here's the code and plots that convinced me.

noiseVector = randn(1,1e5);
[Pxx_noiseOrig, W]=pwelch(noiseVector,hamming(100),[],[],60e6,'centered'); 

filterCoeff = rcosdesign(0.2,10,12);

filteredNoise1 = conv(noiseVector,filterCoeff);
[Pxx_noise1, ~]=pwelch(filteredNoise1,hamming(100),[],[],60e6,'centered'); 


filteredNoise2 = conv(noiseVector, filterCoeff/sum(filterCoeff));
[Pxx_noise2, ~]=pwelch(filteredNoise2,hamming(100),[],[],60e6,'centered'); 

figure('Units','inches')
plot(W/1e6,10*log10([Pxx_noiseOrig, Pxx_noise1, Pxx_noise2]),'Linewidth',2)
grid on
set(gca,'FontSize',12)
legend({'Unfiltered','No scaling','Scaled'},'FontSize',12,'Location','SouthEast')
xlabel('Frequency (MHz)','FontSize',12)
ylabel('Power (dBW)','FontSize',12)
title('Comparison of ouput PSDs','FontSize',14)
get(gcf,'Position');
set(gcf,'Position',[pos(1) pos(2) 5 3])
print('~/Downloads/filteredNoiseComparison','-dpng','-r400')




The plot shows that the original filter had a gain in the passband. After scaling, the noise power in the passband stays the same. My goal was to see how much noise was getting through the filter, so scaled version was the one to use.