digital-communication-0102

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This is the notes of Digital Communication, corresponding to Chapter 2 of Digital Communication by Proakis mainly about the bandpass and lowpass signal representation.

BANDPASS AND LOWPASS SIGNAL REPRESENTATION

Where the Imagenarty Part Comes From?

Assume that we have a real-valued signal $x(t)$, and the spectrum of $x(t)$ is $X(f)$. The spectrum of $x(t)$ is Hermitian symmetric:

$$X(f)=X^*(-f)$$

The summetric property of the spectrum can be shown as:

So we can reconstruct the real-valued signal $x(t)$ as we get the assess to the $X_+(f)$, which is the positive part of the spectrum of $x(t)$. The inverse Fourier transform of $X_+(f)$ is $x_+(t)$:

$$\begin{aligned}x_{+}(t)&=\mathscr{F}^{-1}\left[X_+(f)\right]\\&=\mathscr{F}^{-1}\left[X(f)u_{-1}(f)\right]\\&=x(t)\star\left(\frac12\delta(t)+j\frac1{2\pi t}\right)\\&=\frac12x(t)+\frac j2\widehat{x}(t)\end{aligned}$$

where $\widehat{x}(t)=\dfrac{1}{\pi t}\star x(t)$ is the Hilbert transform of $x(t)$.

Now we define $x_l(t)$ the lowpass equivalent of $x(t)$, whose spectrum is the lowpass version of $X_+(f)$:

$$X_l(f)=2X_+(f+f_0)=2X(f+f_0)u_{-1}(f+f_0)$$

the IFT of which is shown as:

$$\begin{aligned} x_{l}(t)& =\mathscr{F}^{-1}[X_l(f)] \\ &=2x_+(t)e^{-j2\pi f_0t} \\ &=(x(t)+j\widehat{x}(t))e^{-j2\pi f_0t} \\ &=(x(t)\cos2\pi f_0t+\widehat{x}(t)\sin2\pi f_0t) \\ &+j(\widehat{x}(t)\cos2\pi f_0t-x(t)\sin2\pi f_0t) \end{aligned}$$

we have the relation:

$$x(t)=\mathrm{Re}\left[x_l(t)e^{j2\pi f_0t}\right]$$

The real and imaginary parts of $x_l(t)$ are called the in-phase component and the quadrature component of $x(t)$, respectively, and are denoted by $x_i (t)$ and $x_q (t)$. Both $x_i (t)$ and$x_q (t)$ are real-valued lowpass signals, and we have:

$$x_i(t)=x(t)\cos2\pi f_0t+\widehat{x}(t)\sin2\pi f_0t\\x_q(t)=\widehat{x}(t)\cos2\pi f_0t-x(t)\sin2\pi f_0t$$

and

$$x(t)=x_i(t)\cos2\pi f_0t-x_q(t)\sin2\pi f_0t\\\widehat{x}(t)=x_q(t)\cos2\pi f_0t+x_i(t)\sin2\pi f_0t$$

According to the above equations, we can get the structure of the modulator and demodulator of the bandpass signal $x(t)$:

Lowpass Equivalent of a Bandpass System

In the preceding section where we talk about the lowpass equivalent of a bandpass signal, we can also get the lowpass equivalent of a bandpass system, whose impusle response is $h(t)$ with a lowpass equivalent $h_l(t)$:

$$h(t)=\mathrm{Re}\left[h_l(t)e^{j2\pi f_0t}\right]$$

the output of the bandpass system is:

$$Y(f)=X(f)H(f)$$

We can get the similar lowpass equivalent:

$$\begin{aligned} Y_{l}(f)& =2Y(f+f_0)u_{-1}(f+f_0) \\ &=2X(f+f_0)H(f+f_0)u_{-1}(f+f_0) \\ &=\frac12\left[2X(f+f_0)u_{-1}(f+f_0)\right]\left[2H(f+f_0)u_{-1}(f+f_0)\right] \\ &=\frac12X_l(f)H_l(f) \end{aligned}$$

with the time domain representation:

$$y_l(t)=\frac12x_l(t)\star h_l(t)$$

The input-output relation between the lowpass equivalents is very similar to the relation between the bandpass signals, the only difference being that for the lowpass equivalents a factor of $\dfrac{1}{2}$ is introduced.