This section gives a rough introduction into the principles and techniques that can be used to adapt the sample rate of Multirate systems. For a more detailed description please refer to [92, 32, 75].
Basically, there exist two types of signals, namely discrete and continuous- time signals. A continuous-time signal is a time function x(t), which is de- fined for all time t in an interval, usually an infinite interval. Whereas dis- crete time signals only have values at discrete points in time n (denoted by x[n], where n is an integer value).
As shown in Figure 2.3, each sample maintains its voltage level during the
sampling interval T to give the ADC enough time to convert it. This process is
called sample and hold. Since there exists one amplitude level for each sampling
interval, we can sketch each sample amplitude level at its corresponding sam-
pling time instant shown in Figure 2.2, where 14 samples at their sampling time
instants are plotted, each using a vertical bar with a solid circle at its top.
For a given sampling interval T, which is defined as the time span between
two sample points, the sampling rate is therefore given by
f
s¼
1
T
samples per second (Hz):
For example, if a sampling period is T
¼ 125 microseconds, the sampling rate is
determined as f
s¼ 1=125 s ¼ 8,000 samples per second (Hz).
After the analog signal is sampled, we obtain the sampled signal whose
amplitude values are taken at the sampling instants, thus the processor is able
to handle the sample points. Next, we have to ensure that samples are collected
at a rate high enough that the original analog signal can be reconstructed or
recovered later. In other words, we are looking for a minimum sampling rate to
acquire a complete reconstruction of the analog signal from its sampled version.
Analog
filter ADC DSP DAC
Reconstruction filter Analog input Analog output band-limited signal Digital signal Processed digtal signal Output signal
F I G U R E 2 . 1 A digital signal processing scheme.
0 2T 4T
−5 nT
6T 8T 10T 12T
Analog signal/continuous-time signal Signal samples
x (t )
Sampling interval T
5
0
F I G U R E 2 . 2 Display of the analog (continuous) signal and display of digital samples versus the sampling time instants.
Tan: Digital Signaling Processing 0123740908_chap02 Final Proof page 14 21.6.2007 10:57pm Compositor Name: MRaja
14 2 S I G N A L S A M P L I N G A N D Q U A N T I Z A T I O N
Figure 7.4: Display of the analog (continuous) signal and display of digital samples versus the sampling time instants [92].
Chapter 7. Synchronization
Computers or Digital Signal Processors (DSPs) typically work with discrete- time signals. This is due to the fact that continuous-time signals contain infinite number of points. Infinite number of points are not appropriate to be processed by DSPs or computers, since they would require an infinite number of memory and processing power for computation [32]. As shown in Figure 7.4, discrete-time signals result from the sampling of continuous-time signals, but there are also signals that are discrete by nature like the stock market for instance.
Since there exists one amplitude level for each sampling interval, each sample amplitude level can be sketched at its corresponding sampling time instant (see Figure 7.4) where 14 samples at their sampling time instants are plotted, each using a vertical bar with a solid circle at its top.
Moreover, the dynamics of continuous-time signals is represented using dif- ferential equations (derivatives and integrals). The analysis of the dynamic of such continuous-time signals is done using the Laplace transforms, if they are Linear Time Invariant (LTI)3. Furthermore, to do frequency domain analysis Fourier Transforms (FT) is typically used. Whereas, the dynamic of discrete time signals is represented by means of difference equations (differences and sums); analysis is done using Z-transforms, and finally Discrete time Fourier Transforms (DFT) is typically used for frequency domain analysis.
The choice of the domain (time or frequency) for signal analysis depends on the type of application context. For example, looking at continuous-time signals, both Laplace transforms and FT are very closely related. Laplace transforms are more suitable for the analysis of control systems, while FTs are more suitable for communication systems where signal frequency values are important.
For frequency domain analysis, the representation of a digital discrete signal is given in terms of its frequency components. For this purpose, the signal spectrum needs to be developed. The algorithm transforming time domain signal samples into the frequency domain components is known as DFT. The DFT establishes a relationship between the time domain representation and the frequency domain representation [75].
As shown in Figure 7.5, spectral plots are more adequate to display frequency information of digital signals. The figure illustrates the time domain repre- sentation of a sinusoid characterized by 32 samples and sampled at a rate of 8000Hz; the bottom plot shows the frequency domain representation (sig- nal spectrum). The signal spectrum clearly shows that the amplitude peak is located at the frequency of 1000Hz in the calculated spectrum.
Chapter 7. Synchronization 0 5 10 15 20 25 30 −5 0 5 Sample number n x(n) 0 500 1000 1500 2000 2500 3000 3500 4000 0 2 4 6 Frequency (Hz) Signal Spectrum
Figure 7.5: Example of the digital signal and its amplitude spectrum [92]. The signal is sampled at a rate of 8000Hz.
Multirate systemsare complex systems composed of several interconnected sub-components operating with different sampling rates (sampling frequen- cies). The sampling rate defines the number of samples per unit of time (usually seconds). Speech and audio processing are typical application areas. To keep the system synchronized, rising or lowering of the sample rates is required. Basically, the sampling rate of a signal can be manipulated in three different ways [32]:
• Decimation/Downsampling: reduction of the sampling rate by an inte- ger factor M.
• Interpolation/Upsampling: increase of the sampling rate by an integer factor L.
• Resampling: changing the sampling rate by a non-integer factor (L/M). This is a combination of both decimation and interpolation, and de- scribes the interpolation with factor L followed by a decimation with Factor M.