They are the same.
% signal is a sine wave of 2 Hz
nSamples = 10000;
f = 2; % [Hz]
time = linspace(0, 2, nSamples+1);
signal = sin(2*pi*f*time);
% calculate the signal power
signalPower = sum((signal).^2)./nSamples;
% SNR in db is 10log(Psignal/Pnoise)
snrDb = 10; % [dB]
% noise power such that signal power is 10 dB more
% 10 = 10 log (Ps / Pn)
% Pn is variance which for zero mean gaussian noise
% is essentially - square sum of all samples -> divided by numSamples
noiseStd = sqrt(signalPower / 10^(snrDb/10));
noiseMean = 0;
% generate the gaussian white noise
noiseValues = noiseStd*randn(nSamples,1) + noiseMean;
% verify the Signal-to-Noise value
noisePower = sum(noiseValues.^2)/numel(noiseValues);
SNR = 10*log10(signalPower/noisePower);
fprintf(1,"noiseStd: model input: %f simulation output: %f\n", noiseStd, std(noiseValues));
fprintf(1,"SNR: %f\n", SNR);
% compare to awgn
noisedSignal = awgn(signal,10,'measured');
awgnNoiseValues = noisedSignal - signal;
awgnNoisePower = sum(awgnNoiseValues.^2)/numel(awgnNoiseValues);
SNR = 10*log10(signalPower/awgnNoisePower);
fprintf(1,"awgnNoiseStd: simulation output: %f\n", std(awgnNoiseValues));
fprintf(1,"SNR: %f\n", SNR);
