2.1 Dsp00104-sampled time series  (Page 2/10)

While introducing decomposition , I told you that almost everything we will discuss in this series on DSP is based on the premise that every timeseries can be decomposed into a large number of sinusoids, each having its own amplitude and frequency.

I also introduced the concept of composition , where any time series can be created by adding together the correct set of sinusoids, each having itsown amplitude and frequency.

The notion of sampling analog signals

While signal processing can be accomplished in a variety of ways, including analog processors, digital processors, and optical processors, DSP is based onthe notion that signals in nature can be sampled and converted into a series of numbers. The numbers can be fed into some sort of digital device, which canprocess the numbers to achieve some desired objective.

What is meant by sampling ?

To sample a signal means to measure and record its amplitude at a series of points in time. For example, you might record the temperature in your officeevery ten minutes for twenty-four hours. In this case, the actual temperature in your office would be the analog signal. The 144 temperature values that yourecord would be a sampled time series intended to represent that analog signal.

Uniform sampling is most common

Although uniform sampling is not strictly necessary, in DSP, the most common practice is to sample the signal at uniform intervals of time, (such as once every ten minutes, once per second, or one-thousand times per second). This results in a uniform sampling frequency (sampling rate) .

(Most of the discussions in this series of tutorials on DSP will assume a uniform sampling frequency.)

Some problems arise

While sampled data can be used to simulate most of the signal-processing capabilities available with analog devices, the process of sampling doesintroduce some complications that must be dealt with. For the most part, these complications have to do with the relationship between the sampling frequency (in samples per second) and the highest frequency component contained in the signal (in cycles per second).

Stated simply, if the analog signal contains any sinusoidal components whose frequency is greater than half the sampling frequency, then those componentswill appear in the sampled time series at a different frequency. This can result in a variety of problems.

Reconstruction of the analog signal

Theoretically, if the sampling frequency is twice the highest frequency component contained in the analog signal, then the samples can be used inconjunction with an analog filter to reconstruct the original analog signal.

(However, this requires the construction of a perfect analog filter. In practice, the sampling frequency needs to be perhaps five to ten times thehighest frequency component in the analog signal to make it practical to do a good job of reconstructing the analog signal from the samples.)

Reconstruction is not always required

Once the signal has been sampled and converted to digital form, there is often no interest in reconstructing the analog signal from the samples. Whilethis eliminates the difficulty of reconstruction, it doesn't eliminate the potential problems caused by having the sampling frequency be less than twicethe highest frequency component in the signal.