Apply mathematical concepts and derive frequency domain expression of double sideband full carrier wave with AM envelope.

Sure, let's derive the frequency domain expression of a double sideband full carrier (DSB-FC) wave with an AM envelope.

1. First, let's define the time domain expression of a DSB-FC wave. It is given by the product of the message signal and the carrier signal. Mathematically, it can be represented as:

   s(t) = (1+mum(t))cos(2pif_c*t)

   where, s(t) = DSB-FC signal m(t) = message signal mu = modulation index f_c = carrier frequency

2. Now, let's derive the frequency domain expression. The Fourier transform of the DSB-FC signal s(t) can be represented as:

   S(f) = F{s(t)}

   where, S(f) = Fourier transform of s(t) F{.} = Fourier transform operator

3. Applying the Fourier transform to s(t), we get:

   S(f) = F{(1+mum(t))cos(2pif_c*t)}

4. Using the property of Fourier transform, we can split the above expression into two parts:

   S(f) = F{cos(2pif_ct)} + muF{m(t)cos(2pif_ct)}

5. The Fourier transform of cos(2pif_c*t) is a pair of impulses at f = ±f_c. So, the first part of the above expression becomes:

   F{cos(2pif_ct)} = pi[delta(f - f_c) + delta(f + f_c)]

   where, delta(.) = Dirac delta function

6. The second part of the expression is the modulation process, which shifts the spectrum of m(t) to f = ±f_c. So, it becomes:

   muF{m(t)cos(2pif_ct)} = mu[M(f - f_c) + M(f + f_c)]

   where, M(f) = Fourier transform of m(t)

7. Combining the two parts, we get the frequency domain expression of the DSB-FC wave:

   S(f) = pi*[delta(f - f_c) + delta(f + f_c)] + mu*[M(f - f_c) + M(f + f_c)]

This is the frequency domain expression of a double sideband full carrier wave with an AM envelope.