0.16 Lecture 17:modulation, broadcast radio  (Page 2/4)

$$ $$ If the signal is transmitted at a carrier frequency $f_c=\text{600}\text{Mhz}$ , then the $\lambda \approx \frac{3\times {\text{10}}^8\text{m/s}}{6\times {\text{10}}^8\text{Hz}}\approx 0\text{.}5m$ so that an antenna whose length is $\lambda /\text{10}\approx 5$ cm which is much more manageable for the pigeon.

5/ Narrow-band signals

The modulated transduced pigeon sound has a spectrum that is centered about the carrier frequency $f_c$ and has a bandwidth of ${\mathrm{2f}}_m$ where $f_m$ is the maximum frequency of the pigeon sound.

For the pigeon sound we have $f_m$ = 3 kHz and $f_c$ = 600 MHz. Thus, the bandwidth is only ${\text{10}}^{-5}$ of the carrier frequency — an example of a narrowband signal.

An arbitrary narrowband signal can be expressed as

$x(t)=x_c(t)\text{cos}({\mathrm{2\pi f}}_{c}t)+x_s(t)\text{sin}({\mathrm{2\pi f}}_{c}t)$

where $x_c(t)$ and $x_s(t)$ are lowpass time functions. We can expand x(t) as follows

$\begin{array}{}x(t)=\frac{1}{2}(x_c(t)+\frac{1}{j}{x}_s(t))e^{{\mathrm{j2\pi f}}_{c}t}+\frac{1}{2}(x_c(t)-\frac{1}{j}{x}_s(t))e^{-{\mathrm{j2\pi f}}_{c}t},\\ x(t)=R\{(x_c(t)+\frac{1}{j}{x}_s(t))e^{{\mathrm{j2\pi f}}_{c}t}\},\\ x(t)=a(t)\text{cos}({\mathrm{2\pi f}}_{c}t+\varphi (t)),\end{array}$

Where

$a(t)=\sqrt{x_c^2(t)+x_s^2(t)}\begin{array}{cc}& \end{array}\text{and}\begin{array}{cc}& \end{array}\varphi (t)=-{\text{tan}}^{-1}\frac{x_s(t)}{x_c(t)}\text{.}$

An arbitrary narrowband signal can be written as

$x(t)=a(t)\text{Cos}({\mathrm{2\pi f}}_{c}t+\varphi (t))$

Thus, a general narrowband signal contains both amplitude and phase/frequency modulation. In amplitude modulation (AM) $\varphi (t)$ is constant; in phase/frequency modulation (PM or FM) a(t) is constant.

II. AMPLITUDE MODULATION

1/ AM, suppressed carrier

Perhaps the simplest amplitude modulation scheme is the suppressed carrier scheme in which

$x(t)=x_m(t)\times \text{cos}({\mathrm{2\pi f}}_{c}t)$

Therefore, the Fourier transform is

$\begin{array}{}x(f)=x_m(f)\ast F\{\text{cos}({\mathrm{2\pi f}}_{c}t)\},\\ x(f)=x_m(f)\ast \frac{1}{2}(\delta (f-f_c)+\delta (f+f_c)),\\ x(f)=\frac{1}{2}(x_m(f-f_c)+x_m(f+f_c))\text{.}\end{array}$

The Fourier transform of the modulated signal x(t) can be obtained graphically.

The Fourier transform $X_m(f)$ is repeated at $\pm f_c$ .

2/ Demodulation (detection) of AM, suppressed carrier

The original signal $x_m(t)$ can be recovered by modulating the modulated signal and passing the result through a lowpass filter, a process called demodulation or detection.

3/ AM, suppressed carrier radio

A radio communication system that consists of a transmitter and receiver and which uses suppressed carrier AM is shown below.

Therefore,

$\begin{array}{}x(t)=x_m(t)\text{cos}({\mathrm{2\pi f}}_{c}t),y(t)=x(t)\text{cos}({\mathrm{2\pi f}}_{c}t),\text{and}\\ y(t)=x_m(t){\text{cos}}^2({\mathrm{2\pi f}}_{c}t)=\frac{x_m(t)}{2}(1+\text{cos}({\mathrm{4\pi f}}_{c}t))\text{.}\end{array}$

Hence,

$Y(f)=\frac{1}{2}{X}_m(f)+\frac{1}{4}{X}_m(f-{\mathrm{2f}}_c)+\frac{1}{4}{X}_m(f+{\mathrm{2f}}_c)\text{.}$

The spectrum of y(t) involves the spectrum of ${\text{cos}}^2({\mathrm{2\pi f}}_{c}t)$ which can be found by the trigonometric identity or as shown below.

$y(t)=x_m(t){\text{cos}}^2({\mathrm{2\pi f}}_{c}t)=x_m(t)\times \text{cos}({\mathrm{2\pi f}}_{c}t)\times \text{cos}({\mathrm{2\pi f}}_{c}t)$

can be written as

$Y(f)=X_m(f)\ast F\{\text{cos}({\mathrm{2\pi f}}_{c}t)\}\ast F\{\text{cos}({\mathrm{2\pi f}}_{c}t)\}$

The Fourier transform of $F\{\text{cos}({\mathrm{2\pi f}}_{c}t)\}\ast F\{\text{cos}({\mathrm{2\pi f}}_{c}t)\}$ is shown below.

Thus, using Fourier transform properties it is easy to derive trigonometric identities.

Two-minute miniquiz problem

Problem 23-1 — AM, suppressed carrier radio

A slight alternative to the AM suppressed carrier radio system is shown below.

Using an appropriate low pass filter, H(f), to detect $X_m(f)$ , determine the spectrum $Y_m(f)$ .

Solution

We need to convolve the spectrum of the modulated function X(f) with the Fourier transform of $\text{Sin}({\mathrm{2\pi f}}_{c}t)$ .

Thus, the output is zero.

The following reviews the results for suppressed carrier radio.

Suppressed carrier AM requires that the transmitter and receiver be perfectly synchronized. A small difference in frequency of transmitter and receiver oscillators results in a drift in the phase difference between the oscillators which causes variations in the amplitude of the detected signal, called signal strength fading.