Wireless 101: Sampling low-pass signals

Part 2 looks at sampling for band-pass signals.

Uniform Sampling of Signals and Automatic Gain Control
Uniform sampling of both bandlimited lowpass signals and bandpass signals is studied in detail. The conditions for sampling a signal without aliasing are reviewed. Special scenarios relating to the IF frequency and the sampling rate are explained. Signal reconstruction as it relates to the sampling theorem is also discussed. In the context of data conversion, it is imperative to the performance of the modem to control the long-term changes in the received signal strength, thus preserving the overall dynamic range. This must be done without introducing significant clipping if the signal becomes large, or underrepresentation by the data converter if the signal becomes small. Either case would degrade the signal quality and would have adverse effects on the desired SNR. The automatic gain control algorithm discussed in this chapter, one of several ways of implementing gain control, serves to regulate the signal strength in order to preserve optimal performance.

This chapter is divided into four major sections. In Sections 7-1 and 7-2, the uniform sampling theorem is discussed in some detail as it relates to both lowpass and bandpass sampling. That is, uniform sampling as applied to quadrature baseband signaling as well as complex sampling at IF or RF is discussed. The AGC algorithm is considered in Section 7.3. This section also addresses the implication of the peak to average power ratio of a signal and its impact on the AGC. Section 7.4 contains the appendix.

7.1 Sampling of Lowpass Signals
The sampling theorem requires that a lowpass signal be sampled at least at twice the highest frequency component of the analog band-limited signal. This in essence ensures that the spectral replicas that occur due to sampling do not overlap and the original signal can be reconstructed from the samples with theoretically no distortion.

7.1.1 Signal Representation and Sampling
Given a band-limited analog lowpass signal x_a(t) — that is, the highest frequency component of x_a(t) is strictly less than a given upper bound, say B/2 — then x_a(t) can be suitably represented by a discrete signal x(n) made up of uniformly spaced samples collected at a minimum rate of B samples per second. B is known as the Nyquist rate. Inversely, given the discrete samples x(n) sampled at least at the Nyquist rate, the original analog signal x_a(t) can then be reconstructed without loss of information. In the remainder of this section, we will discuss the theory behind lowpass sampling.

The Fourier transform of an analog signal x_a(t) is expressed as

The analog time domain signal can be recovered from X_a(F) via the inverse Fourier transform as:

Next, consider sampling x_a(t) periodically every T_s seconds to obtain the discrete sequence:

The spectrum of x(n) can then be obtained via the Fourier transform of discrete aperiodic signals as:

Similar to the analog case, the discrete signal can then be recovered via the inverse Fourier transform as:

To establish a relationship between the spectra of the analog signal and that of its counterpart discrete signal, we note from (7.2) and (7.3) that:

Note that from (7.6), periodic sampling implies a relationship between analog and discrete frequency f = F/F_s, thus implying that when comparing (7.5) and (7.6) we obtain:

(Click to enlarge)