Problem about gain of the phased.matchedFilter.

I have 2 questions about the phased.matchedFilter.

(1). when you use phased.matchedFilter as follow:

mfilter = phased.MatchedFilter();
[Y,GAIN] = mfilter(X)

I found that the output 'GAIN' is not defined as the SNR gain in the radar handbook, but the total signal power gain. When I normalized the 'Coefficients' of the phased.matchedFilter by its power, the 'GAIN' will be zero, which is demonstrated in the code below.

(2). The pulse compression gain is defined by the waveform's time-bandwidth product. However, in the simulation below, the SNR gain equals the product of sampling rate and bandwidth, which is not consistent with the radar handbook.

% parameter specification
c = 3e8;
fc = 10e9;

PRT = 50e-6;
PRF = 1/PRT;
tw = 10e-6;
bandwidth = 200e6;

fs = 500e6;

% waveform specification
wav = phased.LinearFMWaveform();
wav.SampleRate= fs;
wav.DurationSpecification= 'Pulse width';
wav.PulseWidth= tw;
wav.PRF= PRF;
wav.PRFSelectionInputPort= false;
wav.SweepBandwidth= bandwidth;
wav.SweepDirection= 'Up';
wav.SweepInterval= 'Symmetric';
wav.Envelope= 'Rectangular';
wav.FrequencyOffsetSource= 'Property';
wav.FrequencyOffset= 0;
wav.OutputFormat= 'Pulses';
wav.NumPulses= 1;
wav.PRFOutputPort= false;
wav.CoefficientsOutputPort= false;

% matched filter
mf = phased.MatchedFilter();
coe = getMatchedFilter(wav);
mf.Coefficients = coe./norm(coe);
mf.GainOutputPort = true;

% receiver
rx = phased.Receiver();
rx.Gain = 0;
rx.NoiseFigure = 3;
rx.SampleRate = fs;

% simulation
sig_in = wav();
sig_out = rx(sig_in);

[sig_mf, gain] = mf(sig_out)

sig_mf = 
   0.0044 + 0.0134i
   0.0027 + 0.0083i
  -0.0027 - 0.0083i
  -0.0044 - 0.0135i
  -0.0000 - 0.0001i
   0.0044 + 0.0134i
   0.0028 + 0.0085i
  -0.0027 - 0.0081i
  -0.0044 - 0.0135i
  -0.0001 - 0.0004i
   0.0044 + 0.0133i
   0.0029 + 0.0087i
  -0.0026 - 0.0078i
  -0.0045 - 0.0136i
  -0.0003 - 0.0008i
   0.0043 + 0.0131i
   0.0030 + 0.0091i
  -0.0024 - 0.0074i
  -0.0046 - 0.0138i
  -0.0005 - 0.0014i
   0.0043 + 0.0129i
   0.0032 + 0.0096i
  -0.0022 - 0.0067i
  -0.0046 - 0.0139i
  -0.0007 - 0.0022i
   0.0042 + 0.0125i
   0.0034 + 0.0103i
  -0.0020 - 0.0059i
  -0.0047 - 0.0141i
  -0.0011 - 0.0032i
   0.0040 + 0.0120i
   0.0037 + 0.0110i
  -0.0016 - 0.0049i
  -0.0047 - 0.0141i
  -0.0014 - 0.0043i
   0.0038 + 0.0113i
   0.0039 + 0.0117i
  -0.0012 - 0.0037i
  -0.0047 - 0.0141i
  -0.0019 - 0.0055i
   0.0035 + 0.0105i
   0.0042 + 0.0124i
  -0.0008 - 0.0023i
  -0.0047 - 0.0139i
  -0.0023 - 0.0069i
   0.0032 + 0.0094i
   0.0044 + 0.0131i
  -0.0002 - 0.0007i
  -0.0046 - 0.0136i
  -0.0028 - 0.0083i
   0.0027 + 0.0080i
   0.0046 + 0.0137i
   0.0004 + 0.0011i
  -0.0044 - 0.0129i
  -0.0033 - 0.0097i
   0.0022 + 0.0064i
   0.0048 + 0.0140i
   0.0010 + 0.0030i
  -0.0041 - 0.0120i
  -0.0038 - 0.0111i
   0.0015 + 0.0045i
   0.0048 + 0.0141i
   0.0017 + 0.0051i
  -0.0037 - 0.0107i
  -0.0042 - 0.0123i
   0.0008 + 0.0023i
   0.0048 + 0.0139i
   0.0025 + 0.0072i
  -0.0031 - 0.0090i
  -0.0046 - 0.0133i
  -0.0001 - 0.0001i
   0.0046 + 0.0132i
   0.0032 + 0.0092i
  -0.0024 - 0.0069i
  -0.0048 - 0.0140i
  -0.0009 - 0.0027i
   0.0042 + 0.0121i
   0.0038 + 0.0111i
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  -0.0019 - 0.0054i
   0.0036 + 0.0104i
   0.0044 + 0.0126i
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  -0.0028 - 0.0080i
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   0.0042 + 0.0116i
  -0.0011 - 0.0031i
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  -0.0027 - 0.0074i
   0.0031 + 0.0083i
   0.0050 + 0.0137i
   0.0008 + 0.0021i
  -0.0045 - 0.0121i
  -0.0042 - 0.0114i
   0.0013 + 0.0034i
   0.0052 + 0.0140i
   0.0027 + 0.0073i
  -0.0031 - 0.0084i
  -0.0051 - 0.0137i
  -0.0008 - 0.0021i
   0.0045 + 0.0121i
   0.0042 + 0.0114i
  -0.0012 - 0.0033i
  -0.0052 - 0.0140i
  -0.0028 - 0.0075i
   0.0030 + 0.0082i
   0.0051 + 0.0138i
   0.0010 + 0.0026i
  -0.0044 - 0.0118i
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   0.0010 + 0.0027i
   0.0052 + 0.0138i
   0.0030 + 0.0081i
  -0.0028 - 0.0075i
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  -0.0013 - 0.0034i
   0.0042 + 0.0113i
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   0.0032 + 0.0083i
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  -0.0016 - 0.0042i
   0.0041 + 0.0106i
   0.0049 + 0.0128i
  -0.0001 - 0.0002i
  -0.0050 - 0.0130i
  -0.0040 - 0.0103i
   0.0018 + 0.0046i
   0.0054 + 0.0141i
   0.0026 + 0.0069i
  -0.0033 - 0.0085i
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  -0.0011 - 0.0028i
   0.0044 + 0.0115i
   0.0047 + 0.0122i
  -0.0006 - 0.0015i
  -0.0052 - 0.0134i
  -0.0037 - 0.0096i
   0.0021 + 0.0055i
   0.0055 + 0.0141i
   0.0024 + 0.0061i
  -0.0035 - 0.0091i
  -0.0053 - 0.0136i
  -0.0009 - 0.0022i
   0.0046 + 0.0118i
   0.0047 + 0.0120i
  -0.0007 - 0.0018i
  -0.0053 - 0.0135i
  -0.0037 - 0.0094i
   0.0022 + 0.0057i
   0.0055 + 0.0141i
   0.0024 + 0.0061i
  -0.0035 - 0.0090i
  -0.0054 - 0.0137i
  -0.0010 - 0.0025i
   0.0046 + 0.0116i
   0.0048 + 0.0122i
  -0.0005 - 0.0013i
  -0.0052 - 0.0133i
  -0.0039 - 0.0099i
   0.0020 + 0.0050i
   0.0056 + 0.0141i
   0.0027 + 0.0069i
  -0.0033 - 0.0082i
  -0.0055 - 0.0139i
  -0.0014 - 0.0035i
   0.0043 + 0.0109i
   0.0051 + 0.0128i
  -0.0000 - 0.0000i
  -0.0051 - 0.0128i
  -0.0043 - 0.0109i
   0.0014 + 0.0035i
   0.0055 + 0.0139i
   0.0033 + 0.0084i
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  -0.0022 - 0.0054i
   0.0038 + 0.0095i
   0.0054 + 0.0136i
   0.0009 + 0.0022i
  -0.0047 - 0.0117i
  -0.0049 - 0.0123i
   0.0004 + 0.0011i
   0.0053 + 0.0132i
   0.0041 + 0.0103i
  -0.0017 - 0.0042i
  -0.0056 - 0.0140i
  -0.0032 - 0.0079i
   0.0029 + 0.0071i
   0.0057 + 0.0141i
   0.0021 + 0.0052i
  -0.0039 - 0.0096i
  -0.0055 - 0.0135i
  -0.0009 - 0.0022i
   0.0047 + 0.0116i
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   0.0035 + 0.0087i
   0.0056 + 0.0139i
   0.0015 + 0.0036i
  -0.0043 - 0.0107i
  -0.0053 - 0.0131i
  -0.0004 - 0.0009i
   0.0050 + 0.0122i
   0.0048 + 0.0118i
  -0.0007 - 0.0017i
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   0.0017 + 0.0042i
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   0.0041 + 0.0099i
   0.0057 + 0.0135i
   0.0011 + 0.0026i
  -0.0047 - 0.0111i
  -0.0054 - 0.0129i
  -0.0003 - 0.0008i
   0.0051 + 0.0121i
   0.0051 + 0.0120i
  -0.0004 - 0.0010i
  -0.0055 - 0.0129i
  -0.0047 - 0.0111i
   0.0011 + 0.0027i
   0.0057 + 0.0135i
   0.0042 + 0.0099i
  -0.0018 - 0.0043i
  -0.0059 - 0.0139i
  -0.0037 - 0.0087i
   0.0024 + 0.0057i
   0.0060 + 0.0141i
   0.0032 + 0.0075i
  -0.0030 - 0.0071i
  -0.0060 - 0.0141i
  -0.0026 - 0.0062i
   0.0035 + 0.0083i
   0.0060 + 0.0140i
   0.0021 + 0.0049i
  -0.0040 - 0.0094i
  -0.0059 - 0.0138i
  -0.0015 - 0.0036i
   0.0044 + 0.0103i
   0.0057 + 0.0134i
   0.0010 + 0.0023i
  -0.0048 - 0.0112i
  -0.0055 - 0.0129i
  -0.0005 - 0.0011i
   0.0051 + 0.0119i
   0.0053 + 0.0124i
  -0.0000 - 0.0000i
  -0.0053 - 0.0124i
  -0.0051 - 0.0119i
   0.0005 + 0.0011i
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   0.0048 + 0.0113i
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  -0.0046 - 0.0106i
   0.0013 + 0.0031i
   0.0059 + 0.0136i
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  -0.0017 - 0.0039i
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  -0.0041 - 0.0094i
   0.0021 + 0.0047i
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   0.0022 + 0.0049i
  -0.0040 - 0.0090i
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  -0.0022 - 0.0049i
   0.0040 + 0.0091i
   0.0062 + 0.0139i
   0.0022 + 0.0048i
  -0.0040 - 0.0091i
  -0.0062 - 0.0139i
  -0.0022 - 0.0049i
   0.0040 + 0.0090i
   0.0062 + 0.0139i
   0.0022 + 0.0050i
  -0.0040 - 0.0089i
  -0.0063 - 0.0140i
  -0.0023 - 0.0051i
   0.0039 + 0.0088i
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   0.0024 + 0.0053i
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  -0.0063 - 0.0140i
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   0.0038 + 0.0083i
   0.0063 + 0.0141i
   0.0027 + 0.0059i
  -0.0036 - 0.0080i
  -0.0064 - 0.0141i
  -0.0029 - 0.0063i
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   0.0064 + 0.0141i
   0.0031 + 0.0068i
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   0.0029 + 0.0059i
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  -0.0246 - 0.0139i
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  -0.0199 - 0.0112i
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   0.0001 + 0.0001i
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   0.0163 + 0.0091i
   0.0203 + 0.0114i
   0.0232 + 0.0130i
   0.0249 + 0.0139i
   0.0252 + 0.0141i
   0.0241 + 0.0135i
   0.0217 + 0.0121i
   0.0180 + 0.0101i
   0.0134 + 0.0075i
   0.0080 + 0.0044i
   0.0021 + 0.0012i
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  -0.0098 - 0.0055i
  -0.0151 - 0.0084i
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  -0.0228 - 0.0127i
  -0.0248 - 0.0138i
  -0.0254 - 0.0141i
  -0.0245 - 0.0136i
  -0.0222 - 0.0123i
  -0.0185 - 0.0103i
  -0.0138 - 0.0077i
  -0.0082 - 0.0046i
  -0.0021 - 0.0012i
   0.0041 + 0.0023i
   0.0101 + 0.0056i
   0.0155 + 0.0086i
   0.0200 + 0.0111i
   0.0233 + 0.0129i
   0.0252 + 0.0139i
   0.0256 + 0.0141i
   0.0243 + 0.0134i
   0.0216 + 0.0119i
   0.0175 + 0.0097i
   0.0124 + 0.0068i
   0.0064 + 0.0035i
   0.0000 - 0.0000i

gain = 0

% noise power
noiseSig = sig_out(15000: end);
noisePow = pow2db(mean(noiseSig.*conj(noiseSig)))

noisePow = -117.0121

% signal power before and after matched filter
sigPowBeforeMF = max(db(sig_out));
sigPowAfterMF = max(db(sig_mf));

% measured SNR gain
SNR_before = sigPowBeforeMF - noisePow

SNR_before = 117.0122

SNR_after  = sigPowAfterMF  - noisePow

SNR_after = 154.0018

SNR_gain_measured = SNR_after - SNR_before

SNR_gain_measured = 36.9897

% theoretical SNR gain
SNR_gain_theoretical = pow2db(bandwidth*tw)

SNR_gain_theoretical = 33.0103

SNR_gain = pow2db(fs*tw)

SNR_gain = 36.9897

figure;
plot(db(sig_out));
hold on;
plot(db(sig_mf));
hold off;