VIDEO MODULATION AND TRANSPORT - Digital Video Processing for Engineers - page 158

e j ω c t = cos ω c t + j sin ω c t

Numerically
Controlled
Oscillator

Complex
Multiplier

Digital
Data
Source

QPSK
Modulator

1,0,1.
.

DAC

Real
Part of
Product

I+jQ
symbols

Figure 17.17. Digital upconversion.

For simplicity, assume the output of the modulator is
a complex sinusoid of 1 kHz. The equation for this upconversion
process is:

½ cos ðu carrier t Þþ jsin ðu carrier t Þ ½ cos ð 2 p 1000 t Þþ jsin ð 2 p 1000 t Þ

1
2 ½

¼ cos ððu carrier 2 p 1000 Þ t Þþ cos ððu carrier þ 2 p 1000 Þ t Þ

1
2 ½

ððu carrier p Þ Þ ððu carrier þ p Þ Þ

cos 2 1000 t cos 2 1000 t

1
2 ½

þ ððu carrier þ p Þ Þþ ððu carrier p Þ Þ

j sin 2 1000 t sin 2 1000 t

1
2 ½

þ j sin ððu carrier þ 2 p 1000 Þ t Þþ sin ððu carrier þ 2 p 1000 Þ t Þ

This reduces to:

cos ððu carrier þ 2 p 1000 Þ t Þþ j sin ððu carrier þ 2 p 1000 Þ t Þ

The imaginary portion is discarded. It is not needed because at
carrier frequencies both positive and negative baseband
frequency components can be represented by the spectrum
above and below the carrier frequency.
The final result at the output of the DAC is:

cos ððu carrier þ 2 p 1000 Þ t Þ

Note there is only a signal frequency component above the
carrier frequency. This is because the input is a complex sinusoid
rotating in the positive (counter clockwise) direction

there is no
negative frequency component in this baseband signal.

e