Chapter 2 SIGNALS AND SPECTRA Chapter Objectives: Basic signal properties (DC, RMS, dBm, and power); Fourier transform and spectra; Linear systems and linear distortion; Band limited signals and sampling; Discrete Fourier Transform; Bandwidth of signals. Erhan A. İnce Eeng360 Communication Systems I Department of Electrical and Electronic Engineering Eastern Mediterranean University
Properties of Signals & Noise In communication systems, the received waveform is usually categorized into two parts: Signal: The desired part containing the information. Noise: The undesired part that corrupts the signal Properties of waveforms include: DC value, root-mean-square (rms) value, normalized power, magnitude spectrum, phase spectrum, power spectral density, bandwidth ………………..
Physically Realizable Waveforms Physically realizable waveforms are practical waveforms which can be measured in a laboratory. These waveforms satisfy the following conditions The waveform has significant nonzero values over a composite time interval that is finite. The spectrum of the waveform has significant values over a composite frequency interval that is finite The waveform is a continuous function of time The waveform has a finite peak value The waveform has only real values. That is, at any time, it cannot have a complex value a+jb, where b is nonzero.
Physically Realizable Waveforms Mathematical models that violate some or all of the conditions listed above are often used One main reason is to simplify the mathematical analysis. If we are careful with the mathematical model, the correct result can be obtained when the answer is properly interpreted. Physical Waveform Mathematical Model Waveform The Math model in this example violates the following rules: Continuity Finite duration
Definition: The time average operator is given by, The operator is a linear operator, the average of the sum of two quantities is the same as the sum of their averages:
Periodic Waveforms Definition A waveform w(t) is periodic with period T0 if, w(t) = w(t + T0) for all t where T0 is the smallest positive number A sinusoidal waveform of frequency f0 = 1/T0 Hz is periodic Theorem: If the waveform involved is periodic, the time average operator can be reduced to where T0 is the period of the waveform and a is an arbitrary real constant, which may be taken as zero.
DC Value Definition: The DC (direct “current”) value of a waveform w(t) is given by its time average, ‹w(t)›. Thus, For a physical waveform, we are actually interested in evaluating the DC value only over a finite interval of interest, say, from t1 to t2, so that the dc value is
If we use a mathematical model with a steady-state waveform of infinite extent, then the formula with T can be used
Power Definition. The average power is: Let v(t) denote the voltage across the input of a circuit, and let i(t) denote the current into the terminal, as shown the instantaneous power (incremental work divided by incremental time) associated with the circuit is given by: p(t) = v(t)i(t) the instantaneous power flows into the circuit when p(t) is positive and flows out of the circuit when p(t) is negative. The average power is:
Evaluation of DC value DC Value of this waveform is: p(t) = v(t)i(t) A 120V , 60 Hz fluorescent lamp wired in a high power factor configuration. Assume the voltage and current are both sinusoids and in phase ( unity power factor) Voltage DC Value of this waveform is: Current Instantenous Power p(t) = v(t)i(t)
Evaluation of Power The instantaneous power is: The Average power is: Maximum Power Average Power The maximum power is: Pmax=VI
RMS Value
Normalized Power In the concept of Normalized Power, R is assumed to be 1Ω, although it may be another value in the actual circuit. Another way of expressing this concept is to say that the power is given on a per-ohm basis. It can also be realized that the square root of the normalized power is the rms value. Definition. The average normalized power is given as follows, where w(t) is a voltage or current waveform
Energy and Power Waveforms
Energy and Power Waveforms a waveform can be either an enery or a power signal and not both. If w(t) has finite energy, the power averaged over infinite time is zero. If the power (averaged over infinite time) is finite, the energy if infinite. However, mathematical functions can be found that have both infinite energy and infinite power and, consequently, cannot be classified into either of these two categories. (w(t) = e-t). Physically realizable waveforms are of the energy type. – We can find a finite power for these!!
Decibel A base 10 logarithmic measure of power ratios. The ratio of the power level at the output of a circuit compared with that at the input is often specified by the decibel gain instead of the actual ratio. Decibel measure can be defined in 3 ways Decibel Gain Decibel signal-to-noise ratio Decibel with milli-watt reference or dBm Definition: Decibel Gain of a circuit is:
Decibel Gain If resistive loads are involved,
or
Decibel Signal-to-noise Ratio (SNR) Definition. The decibel signal-to-noise ratio (S/R, SNR) is: Where, Signal Power (S) = And, Noise Power (N) = So, definition is equivalent to
Decibel with Mili watt Reference (dBm) Definition. The decibel power level with respect to 1 mW is: = 30 + 10 log (Actual Power Level (watts)) Here the “m” in the dBm denotes a milliwatt reference. When a 1-W reference level is used, the decibel level is denoted dBW; when a 1-kW reference level is used, the decibel level is denoted dBk. E.g.: If an antenna receives a signal power of 0.3W, what is the received power level in dBm? dBm = 30 + 10xlog(0.3) = 30 + 10x(-0.523)3 = 24.77 dBm
Phasors Definition: A complex number c is said to be a “phasor” if it is used to represent a sinusoidal waveform. That is, where the phasor c = |c|ejc and Re{.} denotes the real part of the complex quantity {.}. The phasor can be written as: