Understand carrier power, output power and dBC?

Question

I have a question regarding the concept of carrier power, output power and power relative to carrier (dBc). Suppose I have a carrier frequency of 4Hz and signal bandwidth = 4 Hz. The signal span is from [2Hz 6Hz].
My indexing starts at 0 for simplicity.

1. I have 7 samples with sampling frequency 7 Hz such that:

   x(n)=[1+1j 2-2j 1+3j 3+j 4 5 2+2j]; % Time domain signal

Representing the signal into frequency domain by taking the FFT with 7 points (each frequency bin is 1 Hz range 0-6 Hz) and normalizing for power,

Nfft=7;

X[n]=fft(x(n),Nfft)/Nfft;  

Therefore,
X[n]=[2.6+0.7j -0.56+0.54j -0.99-0.52j -0.5+0.07j 0.38+0.69j 0.7+0.2j -0.63-0.7j].
The output power in dBm within the Bandwidth from 2Hz to 6Hz (for simplicity we are considering up to only 4 samples excluding 6Hz sample) is:

OutPow=10*log10(sum(abs(X[:,2:5]).^2))+30; % in dBm

Therefore, the signal power is 34.34 dBm. Note: I don't think this is the average power.

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I would like to generate a noise with -20 dBc. Here are two methods to generate noise power with -20dBc in frequency domain.

Method 1 (the idea is Wanted Signal Power (BW 4 Hz) -Noise at the carrier = 20 dBc):

SigPow=34.34 ; %dBm
TargetNoisePower=(34.34-20)=5.66; % Its down to 20 dBc
NoiseCarrierMagnitude=sqrt(1/1000)*10^(5.66/20)=0.0607; % Noise amplitude for carrier
Noise(n)=[0 0 0 0 0.0607 0 0];  % A single sample at frequency domain

Method 2 (The idea is Sample Power at carrier ($f_c=4$ Hz) -Noise at the carrier =20 dBc):

FcPow=27.93 ; %dBm At 4Hz the sample is 0.38+0.69j
TargetNoisePower=(27.93-20)=7.93;  % at the carrier
NoiseCarrierMagnitude=sqrt(1/1000)*10^(7.93/20)=0.079; % Noise amplitude for carrier
Noise(n)=[0 0 0 0 0.079 0 0];  % A single sample at frequency domain

My questions are the following:

1. Which method is right for generating a noise signal in dBc?
2. The channel might not be occupied with signal at the carrier frequency. For example within the signal BW the exact carrier frequency ($f_c=4$ Hz) sample might be 0 or have only some noise (AWGN) sample.
3. When we say signal power within the bandwidth, do we mean the total power within the bandwidth?

Answer 1

Your questions:

1. You might have a carrier frequency, but you don't seem to have a carrier in your signal, so dBc is not a definable concept. You can only generate a noise with a given power in dB of carrier power if there's a carrier.
2. Not a question!
3. Yes

Remarks:

• Your OutPow is not a power, the way you calculate it, but an energy, it's thus also not in dB of milliwatt.
• I think what you're really just after is an SNR in dB. That's just the ratio of average signal power to average noise power.