1. CT1037N Introduction to Communications Signal Representation & Spectral Analysis Er. Saroj Sharan Regmi Lecture 05

2. Issues on Communication. Problems Affecting Communication. Antenna dipole lengths. Sinusoidal Carrier Modulation. Amplitude Modulation & Demodulation. Frequency Modulation & Demodulation. Multiplexing in Frequency, Wavelength or Time Division. Last Lecture: 03 & 04 Analogue Modulation & Multiplexing

3. Signals and Systems, Continuous- and Discrete- Time Systems, Continuous- and Discrete- Time Signals, Fourier Series, One-Sided Amplitude Frequency Spectrum, Two- Sided Amplitude Frequency Spectrum. Today’s Lecture: 05 Signal Representation & Spectral Analysis

4. Signal:Signal: a function of one or more variables that conveys information on the nature of a physical phenomenon. System:System: an entity that manipulates one or more signals to accomplish a function, thereby yielding new signals. System analysis:System analysis: analyze the output signal when input signal and system are given. System synthesis:System synthesis: design the system when input and output signals are given. Signals & Systems

5. Continuous-time System: Continuous-time System: the input and output signals are continuous in time. Continuous Time System

6. Discrete-time System: Discrete-time System: has discrete-time input and output signals. Discrete Time System

7. Continuous-time Signals Discrete-time Signals EvenOdd PeriodicNon-periodic/aperiodic DeterministicRandom EnergyPower Signal Classification

8. x(t): defined for all time t. x[n] : defined only at discrete instants of time. x[n] = x(nT s ), n = 0, ±1, ±2, ±3, … T s : sampling period Continuous- and Discrete- Time Signals (a) Continuous-time signal x(t). (b) Representation of x(t) as a discrete-time signal x[n].

9. Even Signal:Even Signal: Symmetric about the vertical axis. x(-t) = x(t) for all t. Even and Odd Signals Odd Signal:Odd Signal: Asymmetric about the vertical axis. x(-t) = -x(t) for all t.

10. Periodic Signal:Periodic Signal:x(t) = x(t+T), for all t  T = fundamental period  Fundamental frequency, f = 1/T(Hz)  Angular frequency, ω = 2πf (rad/s) Non-Periodic Signal:Non-Periodic Signal:  No value of T satisfies the condition above. Periodic and Non-Periodic Signals

11. Periodic Signal:Periodic Signal: Periodic and Non-Periodic Signals (…2) Non-Periodic Signal:Non-Periodic Signal:

12. What is the fundamental frequency of triangular wave below? Express the fundamental frequency in units of Hz and rad/s? Periodic and Non-Periodic Signals (Example) T = 0.2 secs f = 1/0.2 = 5 Hz ω = 2πf = 2 x 3.14 x 5 = 31.42 rad/s

13. Periodic discrete time signal: x[n] = x[n + N], for integer n Periodic signal Periodic and Non-Periodic Signals For Discrete Time Signals Non-Periodic signal

14. Deterministic signal: there is no uncertainty with respect to its value at any time. Specified function. Random signal: there is uncertainty before it occurs. Deterministic and Random Signals

15. Energy signal:0 < E <  Power signal:0 < P <  Continuous Time Signals Energy and Power Signals Discrete time signals

16. Fourier Series A Fourier series is an expansion of a periodic function in terms of an infinite sum of sines and cosines.  They make use of the orthogonality relationships of the sine and cosine functions. Most useful for a mathematical treatment of signals that are deterministic.  A deterministic waveform can be modelled as a completely specified function of time. For now, we shall only use periodic signals which are necessarily power signals, ie they have:  Finite average power,  Infinite energy.

17. A Periodic Signal An example of a periodic signal: The signal above can be represented by:  a series of sine and cosine terms,  plus a dc term (ie constant value, independent of frequency). T2T t x(t)

18. Frequency Component of A Periodic Signal The Lowest frequency (other than dc) of the sinusoidal components is a frequency, f 1, given by: All other frequencies in the signal will be integer multiples of the fundamental frequency, f 1, and these are called harmonics. Thus:  f 1 is the fundamental frequency or 1 st harmonic,  f 2 is the 2 nd harmonic, even,  f 3 is the 3 rd harmonic, odd,  f 4 is the 4 th harmonic, even,  f n is the n th harmonic.

19. The Periodic Signal f1f1 f2f2 f3f3 The resultant periodic signal: (f 1 + f 2 + f 3 )

20. Bandwidth Issues In any transmission system the bandwidth must be sufficiently large to pass all significant frequencies on the signal. Mathematically, signals contain an infinite number of harmonics suggesting that transmission systems must have infinite bandwidth. In reality this is not so as there are always significant frequencies that can be used to efficiently reproduce the signal, ie some frequency components, usually the weaker ones, can be omitted from the signal and still be able to recognise the signal without noticeable degradation of its quality.

21. Expressing Signals Any periodic signal may be expressed as the sum of sine and/or cosine terms:  This is the ‘sine-cosine’ fourier form. Mathematically: Note:

22. Harmonic Signal Strength The signal strength for a particular harmonic is given by: ie, for the second harmonic term: Note: often the signal may contain only A or B coefficient terms.

23. One-Sided Amplitude Frequency Spectrum Shows the amplitudes C versus frequency. It is not important to show the phase in these plots. Loosely referred to as the ‘Frequency Spectrum’. 0f1f1 f2f2 f3f3 f (Hz) C1C1 C2C2 C3C3 C0C0 CnCn

24. Two-Sided Amplitude Frequency Spectrum Results from the complex representation of the Fourier Series: where we also consider that we have both positive and negative frequencies. It can be shown that the magnitude of is given by:

25. Two-Sided Amplitude Frequency Spectrum (…2) Hence: Thus: 0 f1f1 f2f2 f3f3 -f 3 -f 2 -f 1 f (Hz)

26. Summary Signals and Systems, Continuous- and Discrete- Time Systems, Continuous- and Discrete- Time Signals, Fourier Series, One-Sided Amplitude Frequency Spectrum, Two- Sided Amplitude Frequency Spectrum.