ELX 303 Signals, Spectra, and Processing PDF

ELX 303 – SIGNALS, SPECTRA AND PROCESSING TOPIC 1: Introduction to Signal Processing Specific Learning Outcomes Topic Title Estimated...

ELX 303 – SIGNALS, SPECTRA AND PROCESSING TOPIC 1: Introduction to Signal Processing Specific Learning Outcomes Topic Title Estimated “I SHOULD BE ABLE TO”… time 1 Introduction to Signal Define signal, system and spectra 3 hours Processing Describe properties of a linear system Enumerate the advantages of digital over analog processing BASIC DEFINITIONS A signal x(t) is a set of data or function of time that represents a variable of interest. A signal typically contains information about the nature of a phenomenon. Examples of signals include the atmospheric temperature, humidity, human voice, television images, a dog’s bark, and birdsongs. More generally, a signal may be a function of more than one independent variable (time). For example, pictures are signals that depend on two independent variables (horizontal and vertical positions) and may be regarded as two- dimensional signals. A system is a collection of devices that operate on input signal x(t) (or excitation) to produce an output signal y(t) (or response). A system may also be regarded as a mathematical model of a physical process that relates the input signal to the output signal. Examples of systems include electric circuits, computer programs, the stock market, weather, and the human body. A system may have several mathematical models or representations. The variables in the mathematical model are described as signals, which may be current, voltage, or displacement. In electrical systems, signals are often represented as currents and voltages. In mechanical systems, they are often represented as temperatures, forces, and velocities. In hydraulic systems, signals may be displacements and pressures. Figure 1.1 illustrates the block diagram of a single-input single-output system. We classify the signals that enter the system as input signals, while the signals produced by the system as outputs. For example, we may regard voltages and currents as functions of time in an electric circuit as signals, while the circuit itself is regarded as a system. In engineering systems, signals may carry energy or information. Fig. 1.1. A simple system. A spectrum (plural spectra or spectrums) is a condition that is not limited to a specific set of values but can vary, without steps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors in visible light after passing through a prism. As scientific understanding of light advanced, it came to apply to the entire electromagnetic spectrum. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, or a spectrum of frequencies over a continuous range. The statistical average of a certain signal or sort of signal (including noise) as analyzed in terms of its frequency content, is called its spectrum. Linear Systems Requirements for Linearity A system is called linear if it has two mathematical properties: 1. homogeneity 2. additivity. If you can show that a system has both properties, then you have proven that the system is linear. Likewise, if you can show that a system doesn't have one or both properties, you have proven that it isn't linear. A third property, shift invariance, is not a strict requirement for linearity, but it is a mandatory property for most DSP techniques. When you see the term linear system used in DSP, you should assume it includes shift invariance unless you have reason to believe otherwise. These three properties form the mathematics of how linear system theory is defined and used. Later in this chapter we will look at more intuitive ways of understanding linearity. For now, let's go through these formal mathematical properties. As illustrated in Fig. 1.2, homogeneity means that a change in the input signal's amplitude results in a corresponding change in the output signal's amplitude. In mathematical terms, if an input signal of x[n] results in an output signal of y[n], an input of kx[n] results in an output of ky[n], for any input signal and constant, k.  Fig. 1.2. Definition of homogeneity A simple resistor provides a good example of both homogenous and non-homogeneous systems. If the input to the system is the voltage across the resistor, v(t), and the output from the system is the current through the resistor, i(t) , the system is homogeneous. Ohm's law guarantees this; if the voltage is increased or decreased, there will be a corresponding increase or decrease in the current. Now, consider another system where the input signal is the voltage across the resistor, v(t), but the output signal is the power being dissipated in the resistor, p(t). Since power is proportional to the square of the voltage, if the input signal is increased by a factor of two, the output signal is increase by a factor of four. This system is not homogeneous and therefore cannot be linear. The property of additivity is illustrated in Fig. 1.3. Consider a system where an input of x1[n] produces an output of y1[n]. Further suppose that a different input, x2[n], produces another output, y2[n]. The system is said to be additive, if an input of x1[n] + x2[n] results in an output of y1[n] + y2[n], for all possible input signals. In words, signals added at the input produce signals that are added at the output. Fig. 1.3. Definition of Additivity The important point is that added signals pass through the system without interacting. As an example, think about a telephone conversation with your Aunt Edna and Uncle Bernie. Aunt Edna begins a rather lengthy story about how well her radishes are doing this year. In the background, Uncle Bernie is yelling at the dog for having an accident in his favorite chair. The two voice signals are added and electronically transmitted through the telephone network. Since this system is additive, the sound you hear is the sum of the two voices as they would sound if transmitted individually. You hear Edna and Bernie, not the creature, Ednabernie. A good example of a non-additive circuit is the mixer stage in a radio transmitter. Two signals are present: an audio signal that contains the voice or music, and a carrier wave that can propagate through space when applied to an antenna. The two signals are added and applied to a nonlinear device, such as a pn junction diode. This results in the signals merging to form a third signal, a modulated radio wave capable of carrying the information over great distances. As shown in Fig. 1.4, shift invariance means that a shift in the input signal will result in nothing more than an identical shift in the output signal. In more formal terms, if an input signal of x[n] results in an output of y[n], an input signal of x[n + s] results in an output of y[n + s], for any input signal and any constant, s. Pay particular notice to how the mathematics of this shift is written, it will be used in upcoming chapters. By adding a constant, s, to the independent variable, n, the waveform can be advanced or retarded in the horizontal direction. For example, when s = 2, the signal is shifted left by two samples; when s = -2, the signal is shifted right by two samples. Fig. 1.4. Definition of Shift Invariance The Necessity of Digitization The conventional methods of communication used analog signals for long distance communications, which suffer from many losses such as distortion, interference, and other losses including security breach. In order to overcome these problems, the signals are digitized using different techniques. The digitized signals allow the communication to be more clear and accurate without losses. The following figure indicates the difference between analog and digital signals. The digital signals consist of 1’s and 0’s which indicate High and Low values respectively.  Fig. 1.5. Representation of Signals Advantages of Digital Communication As the signals are digitized, there are many advantages of digital communication over analog communication, such as − 1. The effect of distortion, noise, and interference is much less in digital signals as they are less affected. 2. Digital circuits are more reliable. 3. Digital circuits are easy to design and cheaper than analog circuits. 4. The hardware implementation in digital circuits, is more flexible than analog. 5. The occurrence of cross-talk is very rare in digital communication. 6. The signal is un-altered as the pulse needs a high disturbance to alter its properties, which is very difficult. 7. Signal processing functions such as encryption and compression are employed in digital circuits to maintain the secrecy of the information. 8. The probability of error occurrence is reduced by employing error detecting and error correcting codes. 9. Spread spectrum technique is used to avoid signal jamming. 10. Combining digital signals using Time Division Multiplexing TDM is easier than combining analog signals using Frequency Division Multiplexing FDM. 11. The configuring process of digital signals is easier than analog signals. 12. Digital signals can be saved and retrieved more conveniently than analog signals. 13. Many of the digital circuits have almost common encoding techniques and hence similar devices can be used for a number of purposes. 14. The capacity of the channel is effectively utilized by digital signals. Elements of Digital Communication The elements which form a digital communication system is represented by the following block diagram for the ease of understanding.  Fig. 1.6. Basic Elements of a Digital Communication System Following are the sections of the digital communication system. Source The source can be an analog signal. Example: A Sound signal Input Transducer This is a transducer which takes a physical input and converts it to an electrical signal (Example: microphone). This block also consists of an analog to digital converter where a digital signal is needed for further processes. A digital signal is generally represented by a binary sequence. Source Encoder The source encoder compresses the data into minimum number of bits. This process helps in effective utilization of the bandwidth. It removes the redundant bits unnecessaryexcessbits,i.e.,zeroes. Channel Encoder The channel encoder, does the coding for error correction. During the transmission of the signal, due to the noise in the channel, the signal may get altered and hence to avoid this, the channel encoder adds some redundant bits to the transmitted data. These are the error correcting bits. Digital Modulator The signal to be transmitted is modulated here by a carrier. The signal is also converted to analog from the digital sequence, in order to make it travel through the channel or medium. Channel The channel or a medium, allows the analog signal to transmit from the transmitter end to the receiver end. Digital Demodulator This is the first step at the receiver end. The received signal is demodulated as well as converted again from analog to digital. The signal gets reconstructed here. Channel Decoder The channel decoder, after detecting the sequence, does some error corrections. The distortions which might occur during the transmission, are corrected by adding some redundant bits. This addition of bits helps in the complete recovery of the original signal. Source Decoder The resultant signal is once again digitized by sampling and quantizing so that the pure digital output is obtained without the loss of information. The source decoder recreates the source output. Output Transducer This is the last block which converts the signal into the original physical form, which was at the input of the transmitter. It converts the electrical signal into physical output (Example: loud speaker). Output Signal This is the output which is produced after the whole process. Example − The sound signal received. ACTIVITY NO. 1 Answer the following. 1. In your own understanding, define a. signal b. system c. spectrum 2. What are the properties of a linear systems? Explain in brief. 3. What are the advantages of digital over analog processing? REFERENCES: 1. Sadiko, M N. O. Signals and Systems: A Primer with Matlab. CRC Press. 2016. 2. Tan, L.. Digital Signal Processing: Fundamentals and Applications. Academic Press. 2008. 3. Oppenheim, A. V., Wilsky, A. Signals and Systems. 2nd edition. Prentice Hall. 1996. 4. Orfanidis, S. J. Introduction to Signal Processing, Prentice Hall. 2010. 5. Proakis, J. G., Manolakis, D. G. Digital Signal Processing. 4th edition. Pearson. 2006.