Receiver Hardware Considerations - Fundamentals of Global Positioning System Receivers: A Software Approach - page 124

The above method generates N points of complex data from N points of real
data. The new data may increase the processing load without gaining significant
receiver performance improvement. Therefore, another approach is presented,
which is similar to the above Matlab approach, but generates only N/ 2 points of
complex data. In taking real digitized data the sampling frequency f s 2 . 5 f is

used and the input signal is aliased close to the center of the baseband. Under this
condition, the frequency component X ( N/ 2) should be very small. The following
steps can be taken to obtain complex data:

≈

1. The first step is the same as step 1 (Equation 6.8) in the Matlab approach
to take the FFT of the input signal.
2. The new X 1 (k) can be obtained as

= 0 , 1 , 2 ,..., N X 1 (k) = X(k) for k

2 − 1

( 6 . 12 )

Therefore, only half of the frequency components are kept.
3. The new data in time domain can be obtained as

N
2 −

1

2
N

j 4 πnk
N

x 1 (n) = X 1 (k)e ( 6 . 13 )

k = 0

The final results are N /2 points of complex data in the time domain and they
contain the same information as the N points of real data. These data cover the
same length of time; therefore, the equivalent sampling rate of the complex data
is f s 1 = f s / 2. The argument is reasonable because for complex data the Nyquist

sampling rate is f s 1 = f .

6.14 CHANGE FROM COMPLEX TO REAL DATA

In this section changing complex data to real data will be discussed. The approach
basically reverses the operation in Section 6.13. However, the IF of the down
conversion is very important in this operation. The detail operation depends on
this frequency. One of the common I-Q converter designs is to make the IF at
zero frequency as shown in Figure 6.8. Under this condition, the center frequency
of the input signal is determined by the Doppler shift. For this arrangement the
following steps can be taken:

1. Take the DFT of x ( n ) to generate X ( k ) as shown in Equation (6.8),

N − 1

x(n)e − j 2 πnk X(k) = ( 6 . 14 )

N

n = 0

where x ( n ) is complex, k = 0 , 1 , 2 ,... , N − 1, and n = 0 , 1 , 2 ,... , N − 1.