基本FFT规范化问题

我在用Matlab对信号进行快速傅立叶变换,然后陷入了标准化的困境。具体来说,如何将频谱规范化为dBm的单位。我知道0.316228是正确的标准化因子,但我的问题与如何正确地标准化垃圾箱有关。

我创建了以下程序来提出我的问题。只要把它剪切粘贴到Matlab中,它就会自动运行。请参阅问题。

尤其是,我对如何规范化垃圾箱感到困惑。例如,如果FFT有索引1:end,其中end是偶数,那么当我计算FFT幅度谱时,应该为索引2乘以(2/N)吗:(end/2)?类似地,Nyquist频率(位于索引端/2+1)处的bin是否被规范化为(1/N)?我知道有很多方法可以根据个人的兴趣来正常化。假设我使用的信号(下面的St)是从ADC捕获的电压。

任何反馈都非常感谢。提前谢谢!

%% 1. Create an Example Signal
N = 2^21 ;                   % N = number of points in time-domain signal (St)
St = 1 + rand(N,1,'single'); % St = example broadband signal (e.g. random noise)

% take FFT
Sf = fft(St, N);                    
Sf_mag = (2/N)*abs(Sf(1: N/2 + 1));
Sf_dBm = 20*log10(Sf_mag / 0.316228); % 0.316338 is peak voltage of 1 mW into 50 Ohms

% Q: Are Sf_mag and Sf_dBm normalized correctly? (assume 0.316338 is correct 
%    peak voltage to get 1mW in 50 Ohms)
% Q: Should Sf_mag(fftpoints/2 + 1) = (1/N)*abs(Sf(fftpoints/2 + 1) for correct normalization 
%    of Nyquist frequency? (since Nyquist frequency is not folded in frequency 
%    like the others are)                         

%% 2. Plot Result

% create FFT spectrum x-axis
samplerate = 20e9;  % 20 Gsamples/sec 
fft_xaxis = single(0 : 1 : N/2)';    
fft_xaxis = fft_xaxis * single(samplerate/N); 

semilogx(fft_xaxis, Sf_dBm, 'b-')
xlabel('Frequency (Hz)');
ylabel('FFT Magnitude (dBm)');
title('Spectrum of Signal (Blue) vs Frequency (Hz)');
xlim([1e4 1e10]);
grid on;