Remove "noise" data band from FFT magnitude Plot

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I have data sampled at 128 Hz and I am looking at the frequency content of the data. The matlab FFT function appears to be putting an oscillating band around the output, Figure 1. Is there a way to smooth the FFT plot like what is shown in Figure 2? These aren't plots of the exact same data points but they are pulled from the same data collection. I am not sure if downsampling or filtering the data before the FFT will created a clearner FFT Plot.

Here is my code:

    Fs=128;
  T = 1/Fs;                     % Sample time
  L = n;                     % Length of signal        

NFFT = pow2(nextpow2(L)); % Next power of 2 from length of y
FFTSP=SP-avgSP;
YSP = fft(FFTSP,NFFT)/L;
fSP = Fs/2*linspace(0,1,NFFT/2+1);

figure(1)
subplot(3,1,1)
semilogy(fSP,2*abs(YSP(1:NFFT/2+1)),'Color',[.75 0 0]); 
grid on
xlabel('Frequency (Hz)')
ylabel('|SP|')
xlim([0 30]);
tst2=[stdate2,' to ',eddate2];
title(tst2);
ylim([10^-3 10^1]);