Audio Signal Processing and Coding

2.4 UNIFORM SAMPLING

In all of our subsequent discussions, we will be treating audio signals and associated systems in discrete time. The rules for uniform sampling of analog speech/audio are provided by the sampling theorem [Shan48]. This theorem states that a signal that is strictly bandlimited to a bandwidth of B rad/s can be uniquely represented by its sampled values spaced at uniform intervals that are not more than π/B seconds apart. In other words, if we denote the sampling period as T_s, then the sampling theorem states that . In the frequency domain, and with the sampling frequency defined as ω_s = 2πf_s = 2π/T_s, this condition can be stated as,

Mathematically, the sampling process is represented by time-domain multiplication of the analog signal, x(t), with an impulse train, p(t), as shown in Figure 2.7.

Since multiplication in time is convolution in frequency, the CFT of the sampled signal, x_s(t), corresponds to the CFT of the original analog signal, x(t), convolved with the CFT of the impulse train, p(t). The CFT of the impulses is also a train of uniformly spaced impulses in frequency that are spaced 1/T_s Hz apart. The CFT of the sampled signal is therefore a periodic extension of the CFT of the analog signal as shown in Figure 2.8. In Figure 2.8, the analog signal was considered ...