<< Chapter < Page | Chapter >> Page > |
To continue the investigation, suppose that the carrier phase offset is zero, (i.e., $\Phi =0$ ), but that the frequency offset $\gamma $ is not. Then the spectrum of $x\left(t\right)$ from [link] is
and the lowpass filtering of $x\left(t\right)$ produces
This is shown in [link] . Recognizing this spectrum as a frequency shiftedversion of $w\left(t\right)$ , it can be translated back into the time domain using [link] to give
Instead of recovering the message $w\left(t\right)$ , the frequency offset causes the receiver to recover a low frequencyamplitude modulated version of it. This is bad with even a small carrier frequency offset.While $\mathrm{cos}\left(\Phi \right)$ in [link] is a fixed scaling, $\mathrm{cos}\left(2\pi \gamma t\right)$ in [link] is a time-varying scaling that will alternately recover $m\left(t\right)$ (when $\mathrm{cos}\left(2\pi \gamma t\right)\approx 1$ ) and make recovery impossible (when $\mathrm{cos}\left(2\pi \gamma t\right)\approx 0$ ). Transmitters are typically expected to maintain suitableaccuracy to a nominal carrier frequency setting known to the receiver. Ways of automatically tracking (inevitable) smallfrequency deviations are discussed at length in [link] .
The following code
AM.m
generates a message
$w\left(t\right)$ and
modulates it with a carrier at frequency
${f}_{c}$ .
The demodulation is done with a cosine offrequency
${f}_{c}+\gamma $ and a phase offset of
$\Phi $ .
When
$\gamma =0$ and
$\Phi =0$ , the output
(a lowpass version of the demodulated signal)is nearly identical to the original message,
except for the inevitable delay caused by the linearfilter.
[link] shows four plots:
the message
$w\left(t\right)$ on top, followed by the upconverted signal
$v\left(t\right)=w\left(t\right)cos\left(2\pi {f}_{c}t\right)$ ,
followed in turn by the downconverted signal
$x\left(t\right)$ .
The lowpass filtered versionis shown in the bottom plot; observe that it is nearly identical
to the original message, albeit with a slight delay.
time=.3; Ts=1/10000; % sampling interval and time base
t=Ts:Ts:time; lent=length(t); % define a "time" vectorfc=1000; c=cos(2*pi*fc*t); % define the carrier at freq fc
fm=20; w=5/lent*(1:lent)+cos(2*pi*fm*t); % create "message"v=c.*w; % modulate with carrier
gam=0; phi=0; % freq & phase offset
c2=cos(2*pi*(fc+gam)*t+phi); % create cosine for demodx=v.*c2; % demod received signal
fbe=[0 0.1 0.2 1]; damps=[1 1 0 0]; fl=100; % low pass filter designb=firpm(fl,fbe,damps); % impulse response of LPF
m=2*filter(b,1,x); % LPF the demodulated signal
AM.m
suppressed carrier with (possible) freq and phase offset
(download file)
Using
AM.m
as a starting point,
plot the spectra of
$w\left(t\right)$ ,
$v\left(t\right)$ ,
$x\left(t\right)$ ,
and
$m\left(t\right)$ .
Try different phase offsets $\Phi $ $=$ $[-\pi $ , $-\pi /2$ , $-\pi /3$ , $-\pi /6$ , 0, $\pi /6$ , $\pi /3$ , $\pi /2$ , $\pi ]$ . How well does the recovered message $m\left(t\right)$ match the actual message $w\left(t\right)$ ? For each case, what is the spectrum of $m\left(t\right)$ ?
Notification Switch
Would you like to follow the 'Software receiver design' conversation and receive update notifications?