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Typical transducer Microphone Microprocessor Typical signal generator circuit Signal intensity Time Sound Parameter Signal intensity Time Signal intensity Time Power supply (a) (b) (c) (d) Output Typical FM signal Typical PWM Signal Some of the signals originated from different different signal generato- rs. In (a), the signal is generated from the o/p of a transducer. In (b), the signal has been generated from the o/p of a Microphone, a transduc- er. On the other hand, in (c), the signal has been generated from a typical signal generator circuit, where as in (d), it has been generated from a microprocessor. The sig- nal in (c) and (d) are FM signal and PWM signal respectively. All signals are analog signals. The horizontal axis refers to time axis.

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Electronic filter A signal containing All the frequency components Particular frequency Component (s) of interest For practical use, if we require a signal, which should have a single frequency component or a group of frequency components then we need to fed the original signal, containing all the frequency components to an electronic filter, in order to extract the frequency component (s) of interest. The process is called filtering. The input to the filter is ‘signal’ and the output of the filter is also ‘signal’. However, their frequency contents are not same.

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Time Amplitude Time Amplitude Time Amplitude DC Signal (frequency = 0) Time Amplitude C B A D T (a) (b) (c) (d) Illustration of some of the typical signals as far as their frequency contents are concerned. (a) is a DC signal, with amplitude A, (b) shows a sine and a cosine signal, with peak amplitudes B & frequency f, in each case. In (c) a square wave signal with amplitude C over certain period and amplitude zero over certain period, containing many frequency components, has been shown. (d) is a gate signal, that contains all the frequencies.

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Analog signal Time Sampled signal Sample-1 Sample-2Sample-3 Time Discrete signal Time Discrete levels (a) (b) (c) While an analog signal is continuous in both time and amplitude, a discrete signal is discrete in both time and amplitude. In the discrete domain the signal is defined only by discrete instants of time i.e., the time-variable of the signal takes on only certain values. The value of the signal is sampled and measured at certain intervals in time and each measurement is referred to as a sample. The height of each sample represents the strength of the signal at that sampling instant. The discrete signals are derived from the sampled signal.

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SamplingQuantisation Analog Signal Discrete Signal Sampled Signal Coding Digital Signal The discrete-time signal is generated from the sampled signal by the process of quantisation. The discrete-time signal is then coded in order to obtain the digital signal. This process is called coding. A schematic block diagram as to know how digital signal is generated from the analog signal is shown in this figure.

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Microphone Sound Equivalent electrical signal Amplitude Time A typical analog signal generated from the output of a microphone is shown. The time-amplitude plot or time domain plot of the microphone output only provides amplitude information at various instances. It does not provide any information as far as frequency content of the signal is concerned. One can be interested to know the frequency content of the signal.

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Mathematical treatment (Transformation of time domain signal to frequency domain) FT Time domain signalFrequency domain signal Mathematical treatment (Transformation of time domain signal to time-frequency domain) STFT or WT Time domain signalSignal in time-frequency domain (a) (b) Frequency components 1.Frequency components 2.Time of occurrence of frequency components To know the frequency contents and the time of occurrence of frequency components we need to analyze them in many domains. Once the signal is processed and subsequently plotted, its real face can be opened up. Mathematical treatment is needed in order to obtain frequency information of the signal.

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Amplitude Time A T Tim period Peak amplitude A simple sinusoidal signal (periodic) v(t) is shown in the time domain. The peak amplitude is A and fundamental period is T. It contains only one frequency component.

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Amplitude Time T Tim period A complex periodic signal v(t) is shown in the time domain. The fundamental period is T. It contains many frequency components.

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Amplitude Frequency f A (a) (b) (c) One-sided (Trigonometric form) Frequency Two-sided (Exponential form) DC Component In the frequency domain the sinusoids are plotted as a vertical line and the length of the vertical line represents the peak amplitude of the sinusoids. These are called spectral components, which are discrete in nature. The spectral components are located at an interval of, the fundamental frequency. This signifies that if the duration of time period increases then the spacing between the spectral components decreases. Longer time periods relate to smaller frequency spans.

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Amplitude Time Threshold level Nonperiodic signal (Temperature sensor output) 23 seconds 2 hours This figure shows another example of a nonperiodic signal. The signal is the output of the temperature sensor, which is located at the vicinity of the bearing system of a high-speed spindle machine rotating at a speed more than 15,000 revolutions per minute. The temperature sensor, is a protecting device for the spindle machine. It measures the transient temperature of the bearing system on-line. If the temperature of the bearing system is more than the threshold limit, the spindle has to be stopped, so that no serious damage would occur.

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Frequency Amplitude of Spectral density Figure shows the frequency domain representation of a typical nonperiodic signal. The spectra are continuous. In case of periodic signal that if the duration of time period is increased then the spacing between the spectral components is decreased. If the fundamental period of a periodic signal is increased to infinite, it would become a nonperiodic signal, and in effect, if we take the FT of such signal, the spacing between the spectral components would become infinitesimally small, i.e., continuous.

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Time Amplitude (a) (b) Two typical time domain signals, each containing same frequency components. (a): with frequencies and ; (b) with frequencies and but all are cropping up at different time intervals, and. If we take Fourier transforms of these two signals to know the frequency information we shall get identical results.

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Frequency Amplitude (a) (b) These two figures are the FT of the first and second signals shown in the previous slide. Although, the two signals are exclusively different but they are not distinguished in their frequency domain plots. Therefore, we go for time-frequency distribution. A time-frequency distribution shows what frequencies are present at a given time. STFT and WT are the mathematical tools used to provide time-frequency distribution.

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Time domain function Laplase Trasform (L) Time domain Laplase Inverse Trasform Result Laplace Transform is a useful tool in solving linear, time- invariant differential equations. The technique transforms the original differential equation, which is in time-domain, into the Laplace domain, also called “s” domain. Then it solves the unknowns. Once the job is done, Inverse Laplace Transform is taken in order to give the result. The use of Laplace transformation technique is very popular because some problems are difficult to solve directly. Laplace transform inherits many properties which are of great interest.

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Amplitude Constant-amplitude constant-width pulsed signal Time Amplitude An arbitrary signal (Modulating signal) Time Amplitude Time Amplitude Time PAM Signal PWM Signal ON OFF ON OFF ON OFF Pulse modulation includes Pulse Amplitude Modulation (PAM) and Pulse Width Modulation (PWM). PWM draws special attention as far as automation and control is concerned. One such application is to control the speed of DC motors.

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