0.8 Stuff happens  (Page 4/14)

This ideal system simulation is composed primarily of code recycled from previous chapters.The transformation from a character string to a 4-level $T$ -spaced sequence to an upsampled ( $T/M$ -spaced) $T$ -wide (Hamming) pulse shape filter output sequence mimics pulseshape.m from [link] . This sequence of $T/M$ -spaced pulse filter outputs and its magnitude spectrum are shown in [link] (type plotspec(x,1/M) after running idsys.m ).

Each pulse is 1 time unit long, so successive pulses can be initiated without any overlap.The unit duration of the pulse could be a millisecond (for a pulse frequency of 1 kHz) or a microsecond(for a pulse frequency of 1 MHz). The magnitude spectrum in [link] has little apparent energy outside bandwidth 2 (the meaning of 2 in Hz is dependent on the units of time).

This oversampled waveform is upconverted by multiplication with a sinusoid. This is familiar from AM.m of [link] . The transmitted passband signal and its spectrum(created using plotspec(v,1/M) ) are shown in [link] . The default carrier frequency is fc=20 . Nyquist sampling of the receivedsignal occurs as long as the sample frequency $1/(T/M)=M$ for $T=1$ is twice the highest frequency in the received signal, which will be the carrier frequencyplus the baseband signal bandwidth of approximately 2. Thus, $M$ should be greater than 44 to prevent aliasing of the received signal.This allows reconstruction of the analog received signal at any desired point, which could prove valuable if thetimes at which the samples were taken were not synchronized with the received pulses.

The transmitted signal reaches the receiver portion of the ideal system in [link] . Downconversion is accomplished by multiplying the samples ofthe received signal by an oscillator that (miraculously) has the same frequency and phase as was used in the transmitter.This produces a signal with the spectrum shown in [link] (type plotspec(x2,1/M) after running idsys.m ), a spectrum that has substantial nonzero components (that must be removed)at about twice the carrier frequency.

To suppress the components centered around $\pm 40$ in [link] and to pass the baseband component without alteration (except for possibly a delay), the lowpass filter is designed with acutoff near 10. For $M=100$ , the Nyquist frequency is 50. ( [link] details the use of firpm for filter design.) The frequency response of the resulting FIR filter (from freqz(b) ) is shown in [link] . To make sense of the horizontal axes, observe that the“1” in [link] corresponds to the “50” in [link] . Thus the cutoff between normalized frequencies 0.1 to 0.2corresponds to an unnormalized frequency near 10, as desired.