EE558 - Digital Communications Lecture 2: Review of Signals and Systems Dr. Duy Nguyen
Signals and Systems Input Output signal System signal x [ n ] y [ n ] Signal ◮ Applied to something that conveys information ◮ Represented as a function of one or more independent variables ◮ Continuous-time vs. Discrete-time ◮ Continuous-amplitude vs. Discrete-amplitude System: A transformation or operator that maps a input sequence into an output sequence � � � � y [ n ] = T x [ n ] or y ( t ) = T x ( n ) . 2
Signals Discrete-time signal x [ n ] T ∞ 1 � 2 , � 2 � � � � � � E ∞ = � x [ n ] P = lim � x [ n ] (1) 2 T T →∞ n = −∞ n = − T Some signals have infinite average power, energy or both A signal is called an energy signal if E ∞ < ∞ A signal is called an power signal if 0 < P ∞ < ∞ A signal can be an energy signal, a power signal, or neither type A signal cannot be both an energy signal or a power signal Examples: x [ n ] = 1 , x [ n ] = sin n , x [ n ] = n 3
Some Examples Time shift: x [ n − n 0 ] Time reversal: x [ − n ] Time scaling: x [ an ] Periodic signal with period N : x [ n ] = x [ t + N ] Even signal: x [ − n ] = x [ n ] Odd signal: x [ − n ] = − x [ n ] Exponential signal: x [ n ] = C e an ◮ Real-valued exponential vs Complex exponential ◮ Growing or decaying? ◮ Periodic or aperiodic? Real sinusoidal signal: x [ n ] = A cos( ωn + φ ) 4
Unit Step Function and Unit Impulse Unit step function � 0 , n < 0 u [ n ] = 1 , n > 0 Unit impulse function n � δ [ n ] = u [ n ] − u [ n − 1] , u [ n ] = δ [ m ] m = −∞ 1 1 0 n 0 n Some properties: ∞ � x [ n ] δ [ n − n 0 ] = x [ n 0 ] : sifting property ◮ n −∞ ∞ � ◮ x [ n ] = x [ k ] δ [ n − k ] : signal decomposition k = −∞ 5
Linearity � � Input-output relationship: y i [ n ] = T x i [ n ] A system is linear if ◮ T � � � � ax [ n ] = aT x [ n ] ◮ T � � � � � � x 1 [ n ] + x 2 [ n ] = T x 1 [ n ] + T x 2 [ n ] ◮ or y [ n ] = T � � a 1 x 1 [ n ] + a 2 x 2 [ n ] = a 1 y 1 [ n ] + a 2 y 2 [ n ] . Examples: linear or not Time scaler: y [ n ] = x [2 n ] 1 Amplifier: y [ n ] = 2 x [ n ] + 1 2 n � Accumulator: y [ n ] = x [ k ] 3 k = −∞ Squarer: y [ n ] = x 2 [ n ] 4 6
Causality and Stability Causality: Output only depends on values of the input at only the present and past times Examples: casual or not Time scaler: y [ n ] = x [2 n ] and y [ n ] = x [ n/ 2] 1 � � y [ n ] = sin x [ n ] 2 Stability: Small input lead to responses that do diverge � ≤ B for some B < ∞ − � < ∞ � � � � � x [ n ] → � y [ n ] Examples: stable or not y [ n ] = nx [ n ] 1 y [ n ] = e x [ n ] 2 y [ n ] = y [ n − 1] + x [ n ] 3 7
Time-Invariance Time-invariant system: characteristics of the system are fixed over time � � � � y [ n ] = T x [ n ] − → y [ n − n 0 ] = T x [ n − n 0 ] Examples: Time-invariant or not y [ n ] = sin x [ n ] 1 y [ n ] = nx [ n ] 2 y [ n ] = x [2 n ] 3 Linear time-invariant (LTI) system: good model for many real-life systems Examples: LTI or not n + n 0 1 � y [ n ] = x [ k ] 1 2 n 0 k = n − n 0 8
Response in LTI Systems x [ n ] = δ [ n ] System y [ n ] = h [ n ] Impulse response: Response to a unit impulse Any signal can be expressed as a sum of impulses ∞ � x [ n ] = x [ k ] δ [ n − k ] k = −∞ LTI system: δ [ n − k ] → h [ n − k ] Output signal: ∞ � y [ n ] = x [ k ] h [ n − k ] k = −∞ 9
Convolution Operation ∞ � Convolution operation: y [ n ] = x [ n ] ∗ h [ n ] = x [ k ] h [ n − k ] k = −∞ Commutative: x [ n ] ∗ h [ n ] = h [ n ] ∗ x [ n ] Associative: x [ n ] ∗ ( h 1 [ n ] ∗ h 2 [ n ]) = ( x [ n ] ∗ h 1 [ n ]) ∗ h 2 [ n ] Distributive: x [ n ] ∗ ( h 1 [ n ] + h 2 [ n ]) = x [ n ] ∗ h 1 [ n ] + x [ n ] ∗ h 2 [ n ] Examples: Flip, shift, multiply and add 3 2 2 1 1 1 x [ n ] h [ n ] = y [ n ] ∗ − 2 n − 2 n − 2 n 0 2 4 0 2 0 2 4 10
LTI System Properties and Impulse Response Any LTI system can described by its impulse response Memoryless: h [ n ] = aδ [ n ] Causal: h [ n ] = 0 , ∀ n < 0 ∞ � � < ∞ � � Stable: � h [ n ] n = −∞ 11
Continuous time Signals Unit step function � 0 , t < 0 u ( t ) = 1 , t > 0 Unit impulse function or Dirac delta function � t δ ( t ) = d u ( t ) d t , u ( t ) = δ ( τ )d τ −∞ 1 1 1 t 1 n δ ( t ) = 0 for t � = 0 δ ( t ) in unbounded at t = 0 � ∞ � ∞ δ ( t )d t = 1 and x ( t ) δ ( t − t 0 )d t = x ( t 0 ) : sifting property −∞ −∞ 12
Response in LTI Systems x ( t ) = δ ( t ) System y ( t ) = h ( t ) Impulse response: Response to a unit impulse Any continuous-time signal can be expressed as � ∞ x ( t ) = x ( τ ) δ ( t − τ )d τ −∞ LTI system: δ ( t − τ ) → h ( t − τ ) Output signal: � ∞ x ( τ ) h ( t − τ )d τ � x ( t ) ∗ h ( t ) y ( t ) = −∞ Examples: x ( t ) = e − at u ( t ) , h ( t ) = u ( t ) . Then, y ( t ) = 1 1 − e − at � � . − a 13
Response to Complex Exponentials Input signal: x ( t ) = e st Output signal: � ∞ y ( t ) = e st h ( τ )e − sτ d τ = H ( s )e st −∞ H ( s ) at s : eigenvalue associated with the eigenfunction e st Input signal: x [ n ] = z n Output signal: ∞ h [ k ] z − k = H ( z ) z n y [ n ] = z n � k = −∞ H ( z ) at z : eigenvalue associated with the eigenfunction z n Why is eigenfunction is important? Can any signal be represented as a summation of complex exponentials? 14
Fourier Series I Periodic signal with period T : x ( t ) = x ( t + T ) ω 0 = 2 π/T is called the “angular fundamental frequency” f 0 = 1 /T is called the “fundamental frequency” Harmonically related complex exponentials: Φ k ( t ) = e j kω 0 t Assume a periodic signal x ( t ) can be represented as ∞ � a k e j kω 0 t Synthesis form : x ( t ) = k = −∞ Coefficients a k ’s a k = 1 � x ( t )e − j kω 0 t d t Analysis form: T T 15
Fourier Series II Fourier Analysis using fundamental frequency f 0 = ω 0 / (2 π ) ◮ Synthesis form: ∞ � a k e j k 2 πf 0 t x ( t ) = k = −∞ ◮ Analysis form: � T/ 2 a k = 1 x ( t )e − j k 2 πf 0 t d t T − T/ 2 Parseval’s theorem � T/ 2 ∞ 1 � x 2 ( t )d t = | c k | 2 T − T/ 2 k = −∞ Examples: A periodic square wave 16
Fourier Transform A periodic square wave & Fourier Coefficients � 1 , a k = 2 sin( kω 0 T 1 ) | t | < T 1 x ( t ) = T 1 < | t | < T/ 2 , 0 , kω 0 T Envelop function Ta k = 2 sin ωT 1 � � � ω � ω = kω 0 Fourier series coefficients and their envelop with different values of T with T 1 fixed T → ∞ : Fourier series coefficients approaches the envelope function. 17
Fourier Transform I Aperiodic signal: can be treated as a periodic signal with T → ∞ The envelop function is called the Fourier Transform Derivations of Fourier Transform ◮ Period padding for a aperiodic signal x ( t ) with finite duration x ( t ) ˜ x ( t ) t t 18
Fourier Transform II ◮ Express ˜ x ( t ) using Fourier Series ∞ � a k e j kω 0 t x ( t ) = ˜ k = −∞ where the Fourier Series coefficients are � ∞ a k = 1 x ( t )e j kω 0 t d t = 1 � x ( t )e j kω 0 t d t ˜ T T T −∞ � ∞ x ( t )e − j ωt d t : Analysis Equation of Fourier Define X (j ω ) = −∞ Transform, then a k = 1 T X ( jω ) . Thus, ∞ ∞ 1 1 T X (j kω 0 )e j kω 0 t = � � 2 π X (j kω 0 )e j kω 0 t ω 0 x ( t ) = ˜ k = −∞ k = −∞ 19
Fourier Transform III ◮ As T → ∞ , ω 0 → 0 � ∞ ∞ 1 1 � 2 π X (j kω 0 )e j kω 0 t ω 0 = 2 π X (j ω )e j ωt d ω lim ω 0 → 0 −∞ k = −∞ As ˜ x ( t ) → x ( t ) , Synthesis Equation of Fourier Transform of x ( t ) : � ∞ x ( t ) = 1 X (j ω )e j ωt d ω 2 π −∞ Fourier Transform can be applied to periodic and aperiodic signals. Fourier Series can only be applied to periodic signals Examples: x ( t ) = e − at u ( t ) for a > 0 20
Properties of Fourier Transform I Linearity: if x 1 ( t ) ← → X 1 (j ω ) and x 2 ( t ) ← → X 2 (j ω ) a 1 x 1 ( t ) + a 2 x 2 ( t ) ← → a 1 X 1 (j ω ) + a 2 X 2 (j ω ) → e − j ωt 0 X (j ω ) Time shifting: x ( t − t 0 ) ← Conjugate: x ∗ ( t ) ← → X ∗ ( − j ω ) Differentiation and Integration: d d tx ( t ) ← → j ωX (j ω ) � t → 1 x ( τ )d τ ← j ωX (j ω ) + πX (0) δ ( ω ) −∞ → 1 � j ω � Time scaling: x ( at ) ← | a | X a 21
Properties of Fourier Transform II � ∞ � ∞ � 2 d t = 1 � 2 d ω � � � � Parseval Equality: � x ( t ) � X (j ω ) 2 π −∞ −∞ Duality: Suppose x ( t ) ← → X (j ω ) and y ( t ) ← → Y (j ω ) . If y ( t ) has the shape of X (j ω ) , then Y (j ω ) has the shape of x ( t ) Example: δ ( t ) ← → 1 Convolution: x ( t ) ∗ h ( t ) ← → X (j ω ) H (j ω ) 1 Multiplication: x ( t ) h ( t ) ← → 2 π X (j ω ) ∗ H (j ω ) Fourier Transform can often be denoted as X ( f ) instead of X (j ω ) � ∞ x ( t )e − j2 πft d t X ( f ) = −∞ � ∞ X ( f )e j2 πft d f x ( t ) = −∞ 22
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