ENSC327 Communications Systems 2: Fourier Representations. Jie Liang School of Engineering Science Simon Fraser University

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1 ENSC327 Communications Systems 2: Fourier Representations Jie Liang School of Engineering Science Simon Fraser University 1

2 Outline Chap : Signal Classifications Fourier Transform Dirac Delta Function (Unit Impulse) Fourier Series Bandwidth (Chap will be studied together with Chap. 8) 2

3 Signal Classifications: Deterministic vs Random Deterministic signals: can be modeled as completely specified functions, no uncertainty at all Example: x(t) = sin(a t) Random signals: take random value at any time Example: Noise-corrupted channel output Probability distribution is needed to analyze the signal It is more useful to look at the statistics of the signal: Average, variance Random noise e(t) x(t) y(t): random 3

4 Signal Classifications: Periodic vs Aperiodic Periodic: A signal x(t) is periodic if and only if we can find some constant T0 such that x(t+t0) = x(t), - < t <. t t + T0 Fundamental period: the smallest T0 satisfying the equation above. Aperiodic: Any signal not satisfying the equation is called aperiodic. 4

5 Signal Classifications: Energy Signals vs Power Signals Power and energy of arbitrary signal x(t): Energy: Power: E P = x( t) dt= x( t) dt = lim T lim T T T 1 2T T T 2 x( t) Power is the average amount of energy transferred per unit of time. 2 dt 2 For a periodic signal: What s its energy? P= E= 5

6 Signal Classifications: Energy Signals vs Power Signals A signal is called Energy Signal if its energy is finite 0 < E < (so P = 0) x(t) = 0 at infinity (will be used later) A signal is called Power Signal if its power is finite 0 < P < (so E = ) Periodic signals are power signals, but not energy signals. 6

7 Outline Signal Classifications Fourier Transform Delta Function Fourier Series Bandwidth 7

8 Types of Fourier Series and Transforms Continuoustime signals Discrete-time signals Aperiodic Periodic Aperiodic Periodic 1 2 Continuous-time signals: 1. Aperiodic: 2. Periodic: Discrete-time signals: 3. Aperiodic: 4. Periodic: 3 4 8

9 Fourier Transform (FT) For aperiodic, continuous-time signal: In terms of frequency f: Recall ω = 2 π f 9

10 Amplitude and Phase Spectra G( Important Property: If g(t) is real, then G(f) is conjugate symmetric: G * f) = ( G( f) e jθ( f ) G( f θ ( f phase spectrum f ) = G( f ), or G( f ) = G( f ), θ ( f ) = θ ( f ). Proof: (* is the complex-conjugate operator) ) ) : : amplitude spectrum 10

11 Properties of FT Conjugation rule: g * ( t) G * (-f ) (The notation means that g * ( t) and G * (-f ) are FT pairs) Proof: 11

12 Properties of FT Symmetry: If g(t) is real and even, then G(f) is real and even. If g(t) is real and odd, then G(f) is img. and odd. Proof (first statement only): 12

13 1 Example of Symmetry Unit rectangular function (or gate function): 1, t rect( t) = 0, [ 0.5, ] 0.5, otherwise

14 Example of Symmetry Rectangular Pulse: g(t)= A rect( T t ) G(f) This is a special case of the dilation property (next) 14

15 Properties of Fourier Transform Dilation: Proof: g( at) 1 a f G( a ) (a is a real number) Compress (expand) in time expand (compress) in frequency 15

16 Properties of Fourier Transform Applications of g( at) 1 a G( f a ) g( t) If in addition g(t) is real, G(f) is conjugate symmetric, g( t) Another example: Given Find the FT of g(t)= rect( t) t A rect( ) T sinc( f ) 16

17 Properties of Fourier Transform Duality: Proof: G( t) g( f ) 17

18 Properties of Fourier Transform Duality: Example: G( t) g( f ) g ( t) = A sinc(2wt). Find G( f ). 18

19 Uncertainty Principle of the FT Narrow in time Wide in frequency Wide in time Narrow in frequency 19

20 Properties of Fourier Transform Duality will be used later when we study single sideband (SSB) communications and Hilbert transform 1, sgn(t) = 0, 1, Proof By Duality : t t > = 0, 0, t < 0. 1 jπf 20

21 Properties of Fourier Transform Time shifting (delay): g( t t Time delay only affects the phase spectrum. 0 ) G( f ) e -j2πft 0 Delay g(t) g(t t0) G(f) e -j 2πft 0 G( f ) e -j2πft 0 Proof: 21

22 Properties of Fourier Transform Frequency Shifting: j πf t g( t) e 2 0 G( f f 0 ) Very useful in communications X(f) X(f - f 0) 0 Low freq signal f 0 f 22

23 Properties of Fourier Transform t Example: g( t) = rect( )cos(2πfct ) T 23

24 Properties of Fourier Transform Differentiation: n d x( t) n dt ( ) n j2πf X ( f ) This property is used in FM demodulation 24

25 Properties of Fourier Transform Convolution: the convolution describes the input-output relationship of a linear time-invariant (LTI) system The convolution of two signals is defined as y(t) = The formula is related to the properties of LTI system and impulse response. Note: it is very easy to make mistake about this formula. Please be very careful, as it will appear in the exam. More on this in the end of this lecture. 25

26 Properties of Fourier Transform Convolution property: one of the most useful properties of FT Proof: Let g ( t) G1 ( f ), g2( t) G2( 1 f then g1( τ ) g2( t τ ) dτ G1 ( f ) G2( f ) ), Time domain convolution frequency domain product 26

27 Properties of Fourier Transform Modulation: Proof: g1( t) G1 ( f ), g2( t) G2( f ), g t) g ( t) G( λ) G ( f λ) dλ 1( Time domain product frequency domain convolution 27

28 Rayleigh s Energy Theorem (Parseval s Theorem) g( t) 2 dt = G( f) df 2 Proof: Can calculate the energy in either domain. 28

29 Outline Signal Classifications Fourier Transform Dirac Delta Function (Unit Impulse) Fourier Series Bandwidth 29

30 Dirac Delta Function (Unit Impulse) The Dirac delta function δ(t) is defined to satisfy two relations: δ ( t) = 0 δ ( t) dt for = 1 δ(t) is an even function: δ(-t) = δ(t). The definition implies the sifting property: t 0. g δ(t) ( t 0 ) g(t) g t) δ( t t ) dt ( 0 = δ ( t t ) 0 30 Delta function can be defined by sifting property directly. t 0

31 Dirac Delta Function (Unit Impulse) Since δ(t) is even function, we can rewrite this as Changing the variables, we get the convolution: The convolution of δ(t) with any function is that function itself. This is called the replication property of the delta function. 31

32 Linear and time-invariant system x(t) y(t) Linearity: a system is linear if the input a 1x1( t) + a2x2( t) leads to the output a 1y1( t) + a2 y2( t), where y1, y2 are the output of x1 and y2 respectively. Time-invariant system: a system is time-invariant if the delayed input x( t t ) 0 has the output y t t ), where y(t) is the output of x(t). ( 0 A system is LTI if it s both linear and time-invariant. 32

33 Linear and time-invariant system δ (t) h(t) A linear and time-invariant system is fully characterized by its output to the unit impulse, which is called impulse response, denoted by h(t). The output to any input is the convolution of the input with the impulse response: y ( t) = x( τ) h( t τ) dτ 33

34 Linear and time-invariant system Proof of the convolution expression: We start from the sifting property: This can be viewed as the linear combination of delayed unit impulses. By the properties of LTI, the output of x(t) will be the linear combination of delayed impulse responses: 34

35 Fourier Transform of the delta function By the sifting property, δ ( t) e j2πft dt = The FT of the delta func is This is another example to demonstrate the Uncertainty Principle. 35

36 Applications of Delta Function FT of DC signal: g(t) = 1 G( f ) = δ ( f ). (i.e., DC signal only has 0 frequency component). Proof: Applying duality to g( t) = δ ( t) G(f) = 1 36

37 Applications of Delta Function FT of : (Intuition: a pure complex exponential signal only has one frequency component) Proof: j2π f t j2πf e 0 t δ f f ). e 0 ( 0 37

38 Applications of Delta Function FT of cos 2πf 0 t : FT of sin 2πf 0 t : 38

39 Outline Signal Classifications Fourier Transform Delta Function Fourier Series Bandwidth 39

40 Fourier Series Definition Suppose x(t) is periodic with period T0: Let ω 0= 2π / T0 : jk ot x( t) = X ke ω, t t< t + T k = 1 jkωot X k= x( t) e dt T To o Represent x(t) as the linear combination of fundamental signal and harmonic signals (or basis functions) Xk: Fourier coefficients. Represent the similarity between x(t) and the k-th harmonic signal. 40

41 Fourier Series (cont.) = jk o x( t) X e ω k = k t Example: Find the Fourier series expansion of x( t) = cos( ω t) + sin (2 ω t) Method 1: use the definition jkωot X k= x( t) e dt T T o Method 2: use trigonometric identity and Euler s theorem: o 41

42 Outline Signal Classifications Fourier Transform Delta Function Fourier Series Bandwidth 42

43 Definitions of Bandwidth (Chap 2.3) Bandwidth: A measure of the extent of significant spectral content of the signal in positive frequencies. The definition is not rigorous, because the word significant can have different meanings. For band-limited signal, the bandwidth is well-defined: Low-pass Signals: X(f) X(f) Bandpass signals -W W f 0 fc-w fc f fc+w Bandwidth is W. Bandwidth is 2W. 43

44 Definitions of Bandwidth (Chap 2.3) When the signal is not band-limited: Different definitions exist. Def. 1: Null-to-null bandwidth Null: A frequency at which the spectrum is zero. For low-pass signals: X(f) For Bandpass signals: X(f) 0 f 0 f Bandwidth is half of main lobe width (recall: only pos freq is counted in bandwidth) Bandwidth = main lobe width 44

45 Definitions of Bandwidth (Chap 2.3) Def. 2: 3dB bandwidth Low-pass Signals Bandpass signals X(f) A A/ 2 f X(f) A A/ 2 f 0 0 bandwidth bandwidth X ( f ) 2 drops to 1/2 of the peak value, which corresponds to 3dB difference in the log scale. 10log10 0.5= 3dB 45

46 Definitions of Bandwidth (Chap 2.3) Def. 3: Root Mean-Square (RMS) bandwidth f c G ( f W rms = ( f : center freq. ) = G( f G( f ) ) 2 2 df : f c ) 2 G( f) G( f) 2 df 2 df 1/ 2 Normalized squared spectrum. since G ( f ) df= 1. The RMS bandwidth is the standard deviation of the squared spectrum. 46

47 Definitions of Bandwidth Radio spectrum is a scarce and expensive resource: US license fee: ~ $77 billions / year Communications systems should provide the desired quality of service with the minimum bandwidth. 47