Chapter 2

1. Huseyin Bilgekul Eeng360 Communication Systems I Department of Electrical and Electronic Engineering Eastern Mediterranean University 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.

2. 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 • Properties of waveforms include: • DC value, • Root-mean-square (rms) value, • Normalized power, • Magnitude spectrum, • Phase spectrum, • Power spectral density, • Bandwidth • ………………..

3. 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.

4. 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

5. Time Average Operator • Definition: The time average operatoris given by, • The operator is a linearoperator, • the average of the sum of two quantities is the same as the sum of their averages:

6. Periodic Waveforms • Definition A waveform w(t)is periodicwith period T0if, w(t) = w(t + T0)for all t where T0 is the smallest positive number that satisfies this relationship • A sinusoidal waveform of frequency f0 = 1/T0 Hertz 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 to be zero.

7. 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 t1to t2, so that the dc value is

8. Power • Definition. Let v(t) denote the voltage across a set of circuit terminals, 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 poweris:

9. Evaluation of DC Value • 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) DC Value of this waveform is: Voltage Current Instantenous Power

10. Evaluation of Power The instantaneous power is: The Average power is: The Maximum power is: Pmax=VI

11. RMS Value • Definition: The root-mean-square (rms)value of w(t) is: • Rms value of a sinusoidal: • Theorem: If a load is resistive (i.e., with unity power factor), the average power is: where R is the value of the resistive load.

12. 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 thesquare root of the normalized power is the rms value. Definition. The average normalized poweris given by: Where w(t) is the voltage or current waveform

13. Energy and Power Waveforms • Definition: w(t)is a power waveformif and only if the normalized average power P is finite and nonzero (0 < P < ∞). • Definition: The total normalized energyis • Definition: w(t)is an energy waveformif and only if the total normalized energy is finite and nonzero (0 < E < ∞).

14. Energy and Power Waveforms • If a waveform is classified as either one of these types, it cannot be of the other type. • 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!!

15. 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 • Mill watt Decibel or dBm • Definition:Decibel Gain of a circuit is:

16. Decibel Gain • If resistive loads are involved, Definition of dB may be reduced to, or

17. 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

18. 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

19. Phasors • Definition: A complex number c is said to be a “phasor”if it is used to represent a sinusoidalwaveform. That is, where the phasor c = |c|ejcand Re{.} denotes the real part of the complex quantity {.}. • The phasor can be written as: