By Indzheyan S.G.
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3. 18) Assuming that the noise power density N( f ) can be factored such as N( f ) = Nm ( f ) · Nm∗ ( f ) = |Nm ( f )|2 , the generalized matched filter is equivalent to a cascade of two filters, with transfer ∗ functions Nm1( f ) and NΨ∗ (( ff )) (Fig. 1). e. the noise at the first filter output is white. At the same time, the signal spectrum is altered by the whitening filter. The second filter is the conventional matched filter, which is matched to the distorted signal in the presence of white noise.
7) 2 −∞ This result shows that as long as the phase modulation does not distort the signal envelope, the signal energy is not altered. 8) −∞ The auto-correlation shows how different a signal is compared to its shifted versions as a function of the time shift τ. 3. 3 Complex notation of narrowband signals Signals used in practice are real, however the complex notation offers many advantages particularly in expressing correlation integrals. The matched filter response -the core of the receive processing in coded excitation systems - is a correlation integral and therefore the complex notation is very convenient.
Assuming that the impulse responses are short, the medium’s scattering function γ(t) will be the sum of the impulse responses for all depths. The returned signal will then be the convolution of the scattering function with the excitation signal: r(t) = ψ(t) ∗ γ(t). 21) where τ0 = ri /c is the time instant after the start of transmission (t = 0), at which the signal is being received. The factor k is insignificant for the analysis and will be omitted. The actual imaging situation involves an additional convolution of the transmitted waveform with the transducer impulse response.