ENSC37 Communications Systems : Fourier Representations School o Engineering Science Simon Fraser University
Outline Chap..5: Signal Classiications Fourier Transorm Dirac Delta Function Unit Impulse Fourier Series Bandwidth Chap.6-.9 will be studied together with Chap. 8
Signal Classiication: Deterministic vs Random Deterministic signals: can be modeled as completely speciied unctions, no uncertainty at all Example: xt = sina t Random signals: take a random value at any time Example: Noise-corrupted channel output Probability distribution is needed to analyze the signal It is more useul to look at the statistics o the signal: Average, variance Random noise et xt yt: random 3
Signal Classiication: Periodic vs Aperiodic Periodic: A signal xt is periodic i and only i we can ind some constant T0 such that xt+t0 = xt, -< t <. t t + T0 Fundamental period: the smallest T0 satisying the equation above. Aperiodic: Any signal not satisying the equation is called aperiodic. 4
Signal Classiication: Energy Signals vs Power Signals Power and energy o arbitrary signal xt: Energy: E lim = x t dt = x t dt T T T Power: The average amount o energy per unit o time. For a periodic signal: P = T t0 + T t 0 x t dt P = lim T T T T x t dt What s the energy o a periodic signal? What s the power o an aperiodic and time limited signal? 5
Signal Classiication: Energy Signals vs Power Signals A signal is called an Energy Signal i its energy is inite xt = 0 at ± P = 0 0 < E < A signal is called a Power Signal i its power is inite Periodic Signal are 0 < P < 6
Types o Fourier Series and Transorms Continuoustime signals Discrete-time signals Aperiodic Periodic Aperiodic Periodic Continuous-time signals:. Aperiodic: Fourier transorm. Periodic: Fourier series and Fourier transorm Discrete-time signals: 3 4 3. Aperiodic: Discrete-time Fourier transorm 4. Periodic: Discrete-time Fourier series and Fourier transorm 7
Fourier Transorm FT For aperiodic, continuous-time signal: G ω jωt = = jωt g t e dt, g t G ω e dω ω: continuous angular requency In terms o requency Recall ω = π π G jπt jπt = g t e dt, g t = G e d 8
Amplitude and Phase Spectra G = G e jθ G θ : :amplitude spectrum phase spectrum Important Property: I gt is real, then G is conjugate symmetric: G * = G, or G = G, θ = θ. where * is the complex-conjugate operator. Proo: 9
Properties o FT Conjugation rule: g t G Symmetry: I gt is real and even, then G is real and even. I gt is real and odd, then G is imaginary and odd. g * t G * - 0
Example: Symmetry Rule Find the FT o the Unit rectangular unction or gate unction:, t rect t = 0, [ 0.5, 0.5] otherwise. -0.5 0.5 0.8 0.6 0.4 0. 0-0. -0.4-4 -3 - - 0 3 4 sinc x = π x sin π x very useul in this course
Properties o Fourier Transorm Cont. Dilation or similarity or time scaling/requency scaling : Proo: For g at a G j t a > 0, π g at e dt = For a < 0: a a is a real number g τ e τ jπ a τ d = a G a a. g at e jπt dt = g τ e τ jπ a τ d = a a G a = a G a Note the change o integral range when a<0. Compress expand in time in requency
Example o Dilation Property Find the FT o the Rectangular Pulse o width T and compare with the FT o rectt : gt = A rect T t 3
Uncertainty Principle o the FT Narrow in time Wide in requency Wide in time Narrow in requency 4
Properties o Fourier Transorm Cont. Relection Property: Apply the scaling property : g at a G a to ind the FT o g-t: I in addition gt is real, G is conjugate symmetric, g t G = G *. 5
Properties o Fourier Transorm Cont. Duality: g t G G t g Proo: Since t and are independent variables, by ignoring their physical meanings, we can interchange t and : g - j = G t e πt dt This means that the FT o Gt is g-. 6
Example o Duality Property Apply the duality property to ind the FT o Solution: g t = A sincwt 7
The sign or signum unction sgnt Note: This example is useul later when we study single sideband SSB communications and Hilbert transorm Given sgnt =, 0,, What s the FT o? jπt t > 0, t = 0, t < 0. jπ 8
Properties o Fourier Transorm Cont. Time shiting delay: g t G g t t G 0 e -jπt 0 Time delay only aects the phase spectrum. Frequency shiting: g t G jπ 0 t g t e G 0 Very useul in the study o communication systems modulation: X X - 0 0 Low req signal 0 0 High req signal 9
Example o Frequency Shiting t g t = rect cosπ t T What is the FT o c? This is a modulation technique called amplitude modulation. The eect is that the spectrum Fourier Transorm o rectt/t gets shited to ± cc. 0
Properties o Fourier Transorm Cont. Dierentiation: Proo: n d x t n dt n jπ X Start rom dx t dt j πt x t = X e d jπt de jπt = X d = jπ X e d dt dx t / dt j X π Each derivative operation creates one more term o jπ This property is used in FM demodulation
Convolution Convolution: RecallThe convolution operation describes the input-output relationship o a linear time-invariant LTI system ENSC 380, 383 The convolution o two signals is deined as yt h τ x t τ dτ = x τ h t τ dτ = = x t* h t The ormula is related to the properties o LTI system and impulse response. Note: it is very easy to make mistake about this ormula. Please be very careul, as it may appear in the exam. More on this later.
3 Convolution property one o the most useul properties o FT,, Let G t g G t g = * then G G d t g g t g t g τ τ τ =, Let xt τ τ τ d t g g X G G dtd e t g e g dt e d t g g t j j t j = = = τ τ τ τ τ τ τ π π τ π Proo: Time domain convolution requency domain product Properties o Fourier Transorm Cont.
Properties o Fourier Transorm Cont. Modulation: g t G, g t G, g t g t G λ G λ dλ = G * G Time domain product requency domain convolution 4
Rayleigh s Energy Theorem Parseval s Theorem g t dt = G d Proo: g t dt = g t g * t dt = g t G * e jπt d dt = G * g t e jπt dtd = G * G d = G d We can calculate the total energy o a signal in either domain. 5
Energy Cont. Can we ind the energy o the signal ggtt with the time period tt, tt directly rom it s FT? 6
Dirac Delta Function Unit Impulse The Dirac delta unction δt is deined as a unction satisying two conditions: δ t δ t = 0 or t 0. δ t dt = δt is an even unction: δ-t = δt. The deinition implies the siting property o Delta unction: g t δ t t dt 0 = g t 0 gt δ t t0 t 0 7
Dirac Delta Function Cont. Since δt is even unction, we can rewrite this as g t 0 0 0 = g t δ t t dt = g t δ t t dt. Changing the variables, we get the convolution: g τ δ t τ dτ = g t, or g t δ t = g t. The convolution o δt with any unction is that unction itsel. This is called the replication property o the delta unction. 8
Linear and time-invariant system xt LTI System yt Linearity: a system is linear i the input a x t + ax t leads to the output a y t + a y t, where y, y are the output o x and y respectively. Time-invariant system: a system is time-invariant i the delayed input x t t0 has the output y t t, where yt is the output o xt. 0 A system is LTI i it s both linear and time-invariant. 9
LTI System Cont. δ t LTI System ht A linear and time-invariant system is ully characterized by its output to the unit impulse, which is called impulse response, denoted by ht. The output to any input is the convolution o the input with the impulse response: 30
LTI System Cont. Proo o the convolution expression: We start rom the siting property: x t = x τ δ t τ dτ This can be viewed as the linear combination o delayed unit impulses. By the properties o LTI, δ t τ h t τ the output o xt will be the linear combination o delayed impulse responses: y t = x τ h t τ dτ 3
LTI System Cont. 3
Fourier Transorm o the delta unction 33
Applications o the Unit Impulse Function FT o DC signal: gt = i.e., DC signal only has 0 requency component. G = δ. Proo: We know: g t = δ t G = apply the duality property the above 34
Applications o the Unit Impulse Function Cont. FT o : jπ t jπ 0 t 0. e 0 e δ Intuitively: A pure complex exponential signal only has one requency component Proo: 35
Applications o the Unit Impulse Function Cont. Find the FTs o cos π 0 t and sin π 0 t
Fourier Series Suppose xt is periodic with period T0: Let ω 0 = π / T0 jk ot xt = Xe ω k, t t< t + T k = jkωot X k = x t e dt T To o 0 0 0 Represent xt as the linear combination o undamental signal and harmonic signals or basis unctions XX kk : Fourier coeicients. : 37
Fourier Series cont. jk o xt = Xe ω k k = t Example: Find the Fourier series expansion o Method : Use the deinition xt = cos ω t + sin ω t 0 0 jkωot X k = x t e dt T T o Method : use trigonometric identity and Euler s theorem: jω 0t jω0t cos ω0t = e + e sin ω0t = cos 4 0t = e + e 4 j 0 0 4 ω t j 4 ω t ω o 38
Deinitions o Bandwidth Chap.3 Bandwidth: A measure o the extent o signiicant spectral content o the signal in positive requencies. The deinition is not rigorous, because the word signiicant can have dierent meanings. For band-limited signal, the bandwidth is well-deined: Low-pass Signals: X X Bandpass signals -W W 0 c-w c c+w Bandwidth is W. Bandwidth is W. 39
Deinitions o Bandwidth Cont. When the signal is not band-limited dierent deinitions exist: De. : Null-to-null bandwidth Null: A requency at which the spectrum is zero. For low-pass signals: 0 X For Bandpass signals: 0 X Bandwidth is hal o main lobe width recall: only pos req is counted in bandwidth Bandwidth = main lobe width 40
Deinitions o Bandwidth Cont. De. : 3dB bandwidth Low-pass Signals Bandpass signals X A A/ X A A/ 0 0 bandwidth bandwidth X drops to / o the peak value, which corresponds to 3dB dierence in the log scale. 0log0 0.5 = 3dB 4
Deinitions o Bandwidth Cont. De. 3: Root Mean-Square RMS bandwidth c G W rms = : center req. = G G d : c G G d d / Normalized squared spectrum. since G d =. The RMS bandwidth is the standard deviation o the normalized squared spectrum. 4
Final Note on Bandwidth Radio spectrum is a scarce and expensive resource: US license ee: ~ $77 billions / year Communication system companies try to provide the desired quality o service with minimum bandwidth. 43