Signals and Systems (GPS and Galileo Receiver) Part 3

Sampling

A crucial signal processing operation in a GPS or Galileo software-defined receiver is sampling. In the following we briefly review the sampling process.

Consider the signal x (t). Suppose that we sample this signal at a uniform rate— say once everytmp2D175_thumb[2][2]seconds. Then we obtain an infinite sequence of samples, and we denote this sequence bytmp2D176_thumb[2][2]. where n takes on all integer values. The quantitytmp2D177_thumb[2][2]is called the sampling period, and its reciprocaltmp2D178_thumb[2][2]the sampling rate.

The sampling operation is mathematically described by

tmp2D186_thumb[2][2]

where x(t) is the signal being sampled, andtmp2D190_thumb[2][2]is the sampled signal that consists of a sequence of impulses separated in time bytmp2D191_thumb[2][2]The termtmp2D192_thumb[2][2] represents a delta function positioned at timetmp2D193_thumb[2][2] 

Power spectral densityof a random sequence of pulses.

FIGURE 1.6. Power spectral densitytmp2D188_thumb[2][2]of a random sequence of pulses.

The Fourier transform oftmp2D194_thumb[2][2]is

tmp2D201_thumb[2][2]

Figures 1.7 and 1.8 show the sampling process in the time and frequency domain, respectively.

Figure 1.8 reveals that if the sampling ratetmp2D202_thumb[2][2]is lower that 2B, then the frequency-shifted components of X (f) overlap and the spectrum of the sampled signal is not similar to the spectrum of the original signal x(t). The spectral overlap effect is known as aliasing, and the sampling ratetmp2D203_thumb[2][2]is called the Nyquist rate.

Characterization of Systems

In the continuous-time domain, a system is a functional relationship between the input signal x (t) and the output signal y (t). The input-output relation of a system may be denoted astmp2D208_thumb[2][2]Sampling operation shown in the time domain. Top: Signal x(t). Bottom: Sampled signal

FIGURE 1.7. Sampling operation shown in the time domain. Top: Signal x(t). Bottom: Sampled signaltmp2D210_thumb[2][2]

Figure 1.9 shows a block diagram of a system characterized by a function f and with input signal x (t) and output signal y (t).

By means of the properties of the input-output relationship given in (1.17), we can classify systems as follows:

Linear and nonlinear systems A system is said to be linear if superposition applies. That is, iftmp2D212_thumb[2][2]

Figure 1.9 shows a block diagram of a system characterized by a function f and with input signal x (t) and output signal y (t).

By means of the properties of the input-output relationship given in (1.17), we can classify systems as follows:

tmp2D213_thumb[2][2]

then

tmp2D214_thumb[2][2]

A system in which superposition does not apply is termed a nonlinear system.

Time-invariant and time-varying systems A system is said to be be time-invariant if a time shift in the input results in a corresponding time shift in the output. That is,

tmp2D215_thumb[2][2]

wheretmp2D216_thumb[2][2]is any real number. Systems that do not meet this requirement are called time-varying systems.

tmp2D217_thumb[2][2]Sampling operation shown in the frequency domain. Top: Signal X( f ) with bandwidth B. Bottom:

FIGURE 1.8. Sampling operation shown in the frequency domain. Top: Signal X( f ) with bandwidth B. Bottom:tmp2D219_thumb[2][2]

Causal and noncausal systems A system is said to be causal if its response does not begin before the input is applied, or in other words, the value of the output attmp2D221_thumb[2][2]depends only on the values of x(t) fortmp2D222_thumb[2][2]In mathematical terms, we have

tmp2D226_thumb[2][2]

Noncausal systems do not satisfy the condition given above. Moreover, they do not exist in a real world but can be approximated by the use of time delay.

The classification of continuous-time systems easily carries over to discrete-time systems. Here the input and output signals are sequences, and the system maps the input sequence x(n) into the output sequence y (n).

A simple example of a discrete-time linear system is a system that is a linear combination of the present and two past inputs. Such a system can in general be described by

tmp2D227_thumb[2][2]

and is illustrated in Figure 1.10.

Block diagram representation of a system.

FIGURE 1.9. Block diagram representation of a system.

 Simple linear system with input-output relation

FIGURE 1.10. Simple linear system with input-output relationtmp2D230_thumb[2][2] tmp2D231_thumb[2][2]

Linear Time-Invariant Systems

Let us first consider a continuous-time, linear, time-invariant (LTI) system characterized by an impulse response h(t), which is defined to be the response y(t) from the LTI system to a unit impulsetmp2D236_thumb[2][2]That is,

tmp2D238_thumb[2][2]

The response to the input x (t) is found by convolving x (t) with h (t) in the time domain:

tmp2D239_thumb[2][2]

Convolution is denoted by the *. Sincetmp2D240_thumb[2][2]for causal systems, we can also write y(t) as

tmp2D242_thumb[2][2]

Define the continuous-time exponential signaltmp2D244_thumb[2][2]From (1.19) we then have

tmp2D246_thumb[2][2]

The expression in brackets is the Fourier transform of h(t), which we denote H (f).

Now define the discrete-time exponential sequencetmp2D248_thumb[2][2] We can then characterize a discrete-time LTI system by its frequency response to x(n). By means of the convolution sum formula, a discrete version of (1.20), we obtain the response

tmp2D250_thumb[2][2]

The expression in parentheses is the discrete-time Fourier transform of the impulse response h(n), which we denotetmp2D251_thumb[2][2]

Parameters of a filter.

FIGURE 1.11. Parameters of a filter.

The convolution in the time domain corresponds to the multiplication of Fourier transforms in the frequency domain. Thus, for the system under consideration the Fourier transforms of the input and output signals are related to each other by

tmp2D254_thumb[2][2]

In general, the transfer function H (f) is a complex quantity and can be expressed in magnitude and angle form as

tmp2D255_thumb[2][2]

The quantity | H (f) | is called the amplitude response of the system, and the quantity arg(H (f )) is called the phase response of the system. The magnitude response is often expressed in decibels (dB) using the definition

tmp2D256_thumb[2][2]

We mention in passing that in real systems h (t) is a real-valued function and hence H (f) has conjugate symmetry in the frequency domain, i.e.,tmp2D257_thumb[2][2]

The equations above show that an LTI system acts as a filter. Filters can be classified into lowpass, bandpass, and highpass filters and they are often characterized by stopbands, passband, and half-power (3 dB) bandwidth. These parameters are identified in Figure 1.11 for a bandpass filter.

Magnitude spectrum of bandpass signal s(t) and complex envelope

FIGURE 1.12. Magnitude spectrum of bandpass signal s(t) and complex envelopetmp2D261_thumb[2][2]

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