An Introduction to Signal Processing
Signal processing involves signals and systems.
Today, you have multimedia computers, entertainment systems and digital communications systems and you can be sure that signal processing functions are involved.
When it comes to the meaning of signals, you may be thinking of a signal as carrying information. Usually, that information is describing a physical quantity (e.g. temperature, pressure, or sound waves) that can be managed, stored, or delivered by physical processes.
Examples of signals from physical processes include speech, audio , video or image signals, radar, biomedical, and seismic are just a few of these physical pro
In speech signal that is produced as an acoustic signal, but it can be converted to an electrical signal by a microphone (a transducer), or it can be converted to a string of numbers found in digital audio recording.
Now, let’s turn our attention from signals to systems.
A system can mean many things to a number of different people and is therefore subject to a number of interpretations. For example, a “system” may refer to a large system such as the “Department of Defense”. In terms of signal processing, a system is one in which we can operate, change, record or transmit signals. A simple example would be an audio compact disk (CD), storing and representing a musical signal transcribed as a sequence of digital numbers. Now, a CD player (a system) converts the numbers stored on the disc to an acoustic signal we can hear.
Our use of the term system in signal processing will mean that an input signal will be operated on by a system to produce a new signal at the output of the system.
We will use mathematics to provide more precise statement of signals and systems. Mathematics will be used to show how signals and systems will interact.
How to Represent Signals Mathematically
As discussed earlier, signals carry information. That is, signals are variations represented or encoded patterns of information. They serve to measure or probe real-world physical systems, such as medical technology or wireless communications.
Signals can vary in time, space or both. We’ll initially focus on time since that it is natural to us. One example of a signal is speech which comes from changing pattern of air pressure in the vocal tract. The resulting pattern variation in time is what we call a time waveform.
The above photo shows various time waveforms. The vertical axis is usually the physical quantity such as air pressure and microphone voltage. The horizontal axis represents time. The signals in the above photo show one-dimensional continuous-time signal. Time is normally an independent variable since the signal is a function of time.
Thus, s(t) is a signal function s with an independent time variable t. So at each instant in time we can associate a value s.
Most signals originate as continuous-time signals. However, to increase the flexibility in processing signals, we can use a computer to help process the signal. Unfortunately, storing a continuous-time signal will require an infinite amount of computer memory. As a result, to make it more practical, a continuous-time signal needs to be converted into a discrete-time signal by sampling the signal function at isolated equally spaced points in time. This conversion process will result in a sequence of numbers that is characterized as a function of an index variable n.
Mathematically, where n is an integer and is the sampling period. The reciprocal of is known as the sampling frequency.
The resulting discrete signal from sampling becomes more manageable for the computer or signal-processing hardware.
Normally, we see signals as a function of time. However, there are other signals that do not vary in time. For example, a photo or image does not change in time. Thus, an image can be mathematically described as a function of two spatial variables (e.g. x and y). Here, the variables x and y are two independent variables. We can denote the picture p as a function p(x,y).
The bottom line here is that signals are mathematical functions. Think of functions as abstract symbols of functions. Thus, a speech signal is s(t) or a digitized (or sampled) photo p([m,n]. In this signal processing tutorial we can see that this is a first step to use mathematics to systematically describe signals and systems.
Below are some topics of signal processing videos that I’ve produced so far. More structured articles and videos will follow as well.
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- Digital Signal Processing (DSP) Tutorial: Introduction to Geometric Series and the Discrete-Time Fourier Transform This is an introduction to digital signal processing. Here, we...
- Overview and Introduction to the z-Transform (Polynomial & Rational Functions) We introduce the z-transform bringing polynomials and rational functions to...
- Matlab Examples: Review of Discrete Convolution Sum Using Matlab Here is a review of the discrete convolution sum including...
- z-Transform Tutorial: z-Transform and Inverse z-Transform Examples & Functions In analog or continuous systems the Fourier transform played a...
- Fourier Transform Tutorial and Matlab Resources Here are some useful resources regarding the Fourier Transform as...