Introduction - Signal Processing and Optimization for Transceiver Systems - page 6

f ( t )

(a)

t

0

s (0)

s (1) s (1) f ( t − T )

(b)

s (0) f ( t )

t

0 T 2 T

Figure 1.5 . (a) Impulse response f ( t ) of the prefilter F ( jω ) , and (b) the signal x ( t )
generated from the samples s ( n ) by interpolation with the function f ( t ) .

s ( n )
does not. In practice there is a device called the detector at the receiver (Fig.
1.4), which obtains an estimated constellation symbol s est ( n )fromthequantity

Note that, while s ( n ) belongs to the signal constellation, the quantity

s ( n ) . The probability of symbol error is defined to be the probability that s est ( n )
differs from s ( n ) . The minimization of this probability is another important
optimization problem. Such optimizations and several generalizations will be
discussed in appropriate sections of the topic.

Bandlimiting. In practice the bandwidth allowed for a transceiver is limited. This
bandlimiting is enforced by using lowpass filters at the transmitter and receiver.
These filters can be incorporated as parts of F ( jω )and G ( jω ) . The transmitted
signal x ( t ) therefore occupies a fairly narrow bandwidth of the form

σ<ω<σ, (1 . 12)
andiscalledthe baseband signal. The bandwidth can be a few kHz to MHz,
depending on application. The signal x ( t ) is actually used to modulate a high-
frequency carrier, and the modulated signal is transmitted either wirelessly using
antennas or on wirelines. A discussion of carrier modulation is included in Sec.
2.4. The channel model discussed above is called the baseband model ,asitdoes
not show the carrier explicitly. Similarly the continuous-time system described in
Sec. 1.2 also represents a baseband model.

−

1.3.1 Discrete-time equivalent

We will see in later sections that the problem of designing the digital communi-
cation system of Fig. 1.4 can be reformulated entirely in terms of discrete-time
transfer functions as in Fig. 1.6.