ELEC ENG 4035 Communications IV1 School of Electrical & Electronic Engineering 1 Section 2: Frequency Domain Analysis Contents 2.1 Fourier Series 2.2 Fourier Transforms 2.3 Convolution 2.4 The Sampling Theorem 2.5 The Analytic Signal 2.6 Applications of the Analytic Signal (i) Phasors (ii) Single Sideband Signals

ELEC ENG 4035 Communications IV2 School of Electrical & Electronic Engineering 2 2. Frequency Domain Analysis 2.1 Fourier Series A signal x(t) which has period T can be expressed as:

ELEC ENG 4035 Communications IV3 School of Electrical & Electronic Engineering 3 Notes: (T) means integration over any time interval of length T. We will use f in Hz as the frequency variable. The symbol  will only be used to represent 2  f. Frequency f is a property of the complex exponential e j2  ft and may be positive or negative. Real signals contain both positive and negative frequencies. eg. cos(2  ft) = 0.5e j2  ft + 0.5e –j2  ft.

ELEC ENG 4035 Communications IV4 School of Electrical & Electronic Engineering 4 The number X n is a complex number (dimensions V) and represents a component of frequency n/T Hz, and for a real signal X -n = X n *. This is called the Hermitian property. The fundamental frequency is f o = 1/T corresponding to n = 1. All other frequency components are multiples of this frequency. The amplitude spectrum |X n | of a real signal is an even function of frequency, whereas the phase spectrum arg X n is an odd function.

ELEC ENG 4035 Communications IV5 School of Electrical & Electronic Engineering 5 The average power is obtained by averaging over one period. -2/T -1/T 0 1/T 2/T f -2/T 0 1/T f -1/T 2/T Amplitude SpectrumPhase Spectrum

ELEC ENG 4035 Communications IV6 School of Electrical & Electronic Engineering 6 This is Parseval’s Theorem. Note that P is actually the mean square value.

ELEC ENG 4035 Communications IV7 School of Electrical & Electronic Engineering 7 Example: Rectangular pulse train. -T -  /2 0  /2 T t A

ELEC ENG 4035 Communications IV8 School of Electrical & Electronic Engineering 8

ELEC ENG 4035 Communications IV9 School of Electrical & Electronic Engineering 9 where sinc(x) = sin(  x)/(  x). sinc(x) x

ELEC ENG 4035 Communications IV10 School of Electrical & Electronic Engineering 10 The Fourier series spectrum is: -6/T -5/T -2/T -1/T 0 1/T 2/T 5/T 6/T f -4/T -3/T 3/T 4/T (A  /T) sinc(f  ) XoXo X1X1 X–1X–1 X2X2 X–2X–2

ELEC ENG 4035 Communications IV11 School of Electrical & Electronic Engineering Fourier Transforms The Fourier transform is an extension of Fourier series to non-periodic signals. We will use the notation x(t) ↔ X(f). If x(t) is in volts, the dimensions of X(f) are volt-sec or V/Hz.

ELEC ENG 4035 Communications IV12 School of Electrical & Electronic Engineering 12 Fourier transforms exist for signals of finite energy. This is Rayleigh’s Energy Theorem.

ELEC ENG 4035 Communications IV13 School of Electrical & Electronic Engineering 13 Note that G xx (f) = |X(f)| 2 is also called the energy spectral density of the finite energy signal. Example: Rectangular pulse -  /2  /2 t -2/  -1/  1/  2/  f A A 

ELEC ENG 4035 Communications IV14 School of Electrical & Electronic Engineering 14 Example: Exponential pulse 0 t 0 f +90 o 1/a -90 o 1

ELEC ENG 4035 Communications IV15 School of Electrical & Electronic Engineering 15 Make sure that you know how to deal with: Time scaling Frequency scaling Time shifts Frequency shifts Time inversion Differentiation Integration Multiplication by t See the Fourier transform sheet provided

ELEC ENG 4035 Communications IV16 School of Electrical & Electronic Engineering 16 For periodic signals we have: Exercise: Prove that X n = (1/T) G(n/T).

ELEC ENG 4035 Communications IV17 School of Electrical & Electronic Engineering Convolution If we have V(f) = X(f)Y(f), what is v(t)?

ELEC ENG 4035 Communications IV18 School of Electrical & Electronic Engineering 18 This is called the convolution of x(t) and y(t). We can also define convolution in the frequency domain. Exercise: Prove

ELEC ENG 4035 Communications IV19 School of Electrical & Electronic Engineering The Sampling Theorem If a signal is sampled at intervals of T = 1/f s we have: We can recover x(t) from x s (t) by low pass filtering if the bandwidth of x(t) is less than 1/2T (ie. less than one half the sampling rate).

ELEC ENG 4035 Communications IV20 School of Electrical & Electronic Engineering 20 -W 0 W f X(f) A X s (f) -f s -W f s +W -W 0 W f s -W f s +W f …. A/T

ELEC ENG 4035 Communications IV21 School of Electrical & Electronic Engineering 21 If we sample at lower than the required rate of f s = 2W, we get aliasing. This is where signals at frequencies greater than the Nyquist frequency f s /2, reappear at frequencies mirror imaged about f s /2. For instance if the sampling rate is 10 kHz, a 6.5 kHz signal will appear as a 3.5 kHz signal when we try to recover the signal. To avoid this, frequencies higher than f s /2 must be removed by an anti-aliasing filter before sampling.

ELEC ENG 4035 Communications IV22 School of Electrical & Electronic Engineering 22 Frequency 7500 Hz Frequency 2500 Hz Sampling frequency Hz

ELEC ENG 4035 Communications IV23 School of Electrical & Electronic Engineering The Analytic Signal With a real signal x(t) we have X(–f) = X * (f), so the negative frequency part is redundant. As for sinewaves, it is convenient to deal with signals which contain only positive frequencies. A cos(2  f o t +  ) = Re{A e j(2  f o t +  ) } (Real signal) (Analytic signal) The analytic signal is also called the pre-envelope.

ELEC ENG 4035 Communications IV24 School of Electrical & Electronic Engineering 24 To obtain the analytic signal, we simply discard the negative frequency part and double the positive frequency part. For a real signal x(t), we designate the analytic signal as x + (t). X(f) X + (f)  22 0 f

ELEC ENG 4035 Communications IV25 School of Electrical & Electronic Engineering 25 Example:

ELEC ENG 4035 Communications IV26 School of Electrical & Electronic Engineering 26

ELEC ENG 4035 Communications IV27 School of Electrical & Electronic Engineering 27 The real part of the analytic signal is the original signal, the imaginary part is called the Hilbert transform of x(t) and is denoted.

ELEC ENG 4035 Communications IV28 School of Electrical & Electronic Engineering 28 We recognise this as a filtering operation with a filter: H(f) = –j sgn(f). x(t) H(f) x(t) ^ |H(f)| = o -90 o 0 f

ELEC ENG 4035 Communications IV29 School of Electrical & Electronic Engineering 29 The impulse response of the Hilbert transformer is: The Hilbert transformer is non-causal. Bandlimiting removes the infinity at t = 0. h(t) 0 t

ELEC ENG 4035 Communications IV30 School of Electrical & Electronic Engineering Applications of the Analytic Signal (i) Phasors With a sinusoid x(t) = A cos(2  f o t +  ) the analytic signal is x + (t) = A e j(2  f o t +  ). If we factor out the e j2  f o t term, the remaining factor Ae j  is the phasor representing x(t). An important aspect of the phasor is the reference frequency f o (specified separately).

ELEC ENG 4035 Communications IV31 School of Electrical & Electronic Engineering 31 The reference frequency f o is arbitrary, but for a sinewave A cos(2  f s t+  ) we usually choose it as the frequency of the sinewave, since then the phasor is a constant. (f o = f s ) Imag Real A 

ELEC ENG 4035 Communications IV32 School of Electrical & Electronic Engineering 32 Phasors are of most use when we consider narrowband signals. These are signals which have their frequency components concentrated near some frequency f o. X(f)  -f o 0 f o f

ELEC ENG 4035 Communications IV33 School of Electrical & Electronic Engineering 33 The phasor is denoted and is simply a frequency down-shifted version of the analytic signal. It contains all the information about x(t) except the carrier frequency f o. X + (f) Analytic signal X(f) Phasor 2  0 f o f 0 f ~

ELEC ENG 4035 Communications IV34 School of Electrical & Electronic Engineering 34 Envelope Relative phase Frequency deviation

ELEC ENG 4035 Communications IV35 School of Electrical & Electronic Engineering 35 (ii) Single Sideband Signals A single sideband (SSB) signal is one in which the positive frequency components of a baseband signal x(t) are translated up by f o and the negative frequency components down by f o. To determine an expression describing how the SSB signal y(t) is related to the baseband signal x(t), we first consider the relation between the respective analytic signals.

ELEC ENG 4035 Communications IV36 School of Electrical & Electronic Engineering 36 X(f) Y(f)  0 f -f o 0 f o f X + (f) Y + (f) 2  0 f -f o 0 f o f

ELEC ENG 4035 Communications IV37 School of Electrical & Electronic Engineering 37 Exercise: For x(t) = 4 sinc(2t) and f o = 10 Hz, calculate the spectrum of and hence that of y(t) from the expression above. Sketch the spectra obtained in each case.

ELEC ENG 4035 Communications IV38 School of Electrical & Electronic Engineering 38 Exercises: You are expected to attempt the following exercises in Proakis & Salehi. Completion of these exercises is part of the course. Solutions will be available later (f = 0 should be f > 0)