EE 224 Signals and Systems I Review 1/10

EE 224 Signals and Systems I Review 1/10

Class Contents Signals and Systems Continuous-Time and Discrete-Time Time-Domain and Frequency Domain (all these dimensions are tightly coupled) SIGNALS SYSTEMS TIME CT signals DT signals CT LTI: IR, Conv DT LTI: IR, Conv FREQ CTFS, CTFT, Laplace (sampling, modulation) DTFS, DTFT, Z CT LTI: FR, Multipl. DT LTI: FR, Multipl. 2/10

Signals in Time Domain Properties of signals: periodicity even/odd symmetry signal energy/power Time transformations of signals: shift reversal scaling combinations of these 3/10 a

Signals in Time Domain CT basic signals Sinusoids and phasor addtiion Complex exponentials e jωt Real exponentials e at δ(t) and u(t) x(t) δ(t τ) = x(τ)δ(t τ) x(t) δ(t τ)dt = x(τ) x(t) δ(t t 0 ) = x(t t 0 ) 3/10 b

Signals in Time Domain DT basic signals Unit Impulse δ[n] Exponential signals, especially e jωn Sinusoids cos(ωn + θ) Unit step function u[n] 3/10 c

Signals in FREQ domain: Fourier Analysis FS FT 1 1/5 7f 0 Spectrum 5f 0 1/3 3f 0 1/7 f... 1 7 sin(2π7f 0t) 1 5 sin(2π5f 0t) 1 3 sin(2π3f 0t) ω f 0 sin(2π f 0 t) e jωt t t 4/10 a

Signals in FREQ domain: Fourier Analysis CT DT Periodic (FS) x(t) = CTFS a k = 1 T 0 DTFS k= <T 0 > a k e jkω 0t x(t)e jkω 0t dt x[n] = N 0 1 k=0 a k e jkω 0n a k = 1 N 0 <N 0 > x[n]e jkω 0n Non-Periodic (FT) x(t) = 1 2π x[n] = 1 2π CTFT X(ω)ejωt dω X(ω) = x(t)e jωt dt DTFT π π X(Ω)ejΩn dω X(Ω) = k= x[n]e jωn 4/10 b

Signals in FREQ domain: Fourier Analysis Properties of CT Fourier Transform x(t) X(ω) Time Frequency Duality X(t) 2πx( ω) Linear ax(t) + by(t) ax(ω) + by (ω) Conv. x(t) y(t) X(ω) Y (ω) 1 Modu. x(t) y(t) 2π X(ω) Y (ω) Int. x(t) u(t) = t x(τ)dτ 1 jω X(ω) + πx(0) δ(ω) Diff x(t) δ (t) = d dtx(t) jω X(ω) derivative ω tx(t) j d dω X(ω) time shift x(t τ) X(ω) e jωτ freq shift e jω0t x(t) X(ω ω 0 ) 1 scaling x(at) a X ( ) ω a time reversal x( t) X( ω) conjugation x (t) X ( ω) Parseval x(t) y 1 (t)dt X(ω) Y (ω)dω F 2π 4/10 c

Signals in FREQ domain: Fourier Analysis FT of periodic signals Define one period of x(t) as x T0 (t), which is equal to x(t) over one period (any period), and zero outside that period. Then x(t) = x T0 (t) i= δ(t it 0). As a result, the spectrum is an envelop shape sampled by deltas: X(jω) = X T0 (jω) k= 2π δ(ω k 2π ). T 0 T 0 This gives the envelop of the spectrum This gives the location of the Deltas 4/10 d

Signals in FREQ domain: Sampling Impulse Train Sampling time frequency 5/10 a

Signals in FREQ domain: Sampling If x(t) is bandlimited so that Nyquist-Shannon Sampling Theorem X(jω) = 0 for ω > ω M, then x(t) can be uniquely determined by its samples {x(nt )} if ω s 2π T > 2ω M. 5/10 b

Signals in FREQ domain: Sampling Aliasing when sampling below Nyquist rate no aliasing aliasing 0 ωs ω s ω 2 ω 5/10 c

Signals in FREQ domain: Sampling Aliasing when sampling below Nyquist rate spectrum of cosine 2 2 f spectrum of impulse train... 3 0 3 spectrum of sampled signal... 2 1 0 1 2 5/10 d

Signals in FREQ domain: Sampling Anti-Aliasing Filter analog LPF, ω s /2 anti-aliasing filter sample and hold ADC digital without anti-aliasing filter ω with anti-aliasing filter ω 5/10 e

Signals in FREQ domain: Sampling Reconstruction Ideal reconstruction Zero-order hold Linear interpolation h 0 (t nt ) h 1 (t nt ) 5/10 f

Signals in FREQ domain: Modulation DSB AM Modulation 6/10 a

Signals in FREQ domain: Modulation DSB AM Demodulation Other modulation: AM with carrier and envelop detector; SSB; Frequency-division multiplexing; Pulse amplitude modulation; PM, FM. 6/10 b

Systems in TIME domain System properties: Memory Invertibility Causality Stability Time Invariance Linearity Properties of LTI Systems Commutative Property, Distributive Property, Associative Property, System Memory, Invertibility, Causality, Stability, Unit Step Response 7/10

Systems in TIME domain Impulse Response input DT LTI systems DT LTI System h[n] output Convolution y[n] = x[n] h[n] = k= x[k]h[n k] Properties of convolution: linear, commutative, associative Step response: response to u[n] 8/10 a

Systems in TIME domain DT LTI systems Simple convolutions x[n] δ[n] = x[n] x[n] δ[n n 0 ] = x[n n 0 ] x[n] u[n] = n k= x[k] Graphical approach of doing convolution Fix one signal, say x[k] Flip the other signal h[k] and shift right by n to obtain h[n k] (as a function of k) Multiply x[k] and h[n k] Sum over k Repeat for another n Can also do convolution using Fourier transform 8/10 b

Systems in TIME domain Causal LTI Systems Described by Difference Equations Example N k=0 DT LTI systems a k y[n k] = F [0] = 0, F [1] = 1 Important assumption for LTI and causality: initially at rest The complete solution can be written as the sum of two parts: homogeneous solution (natural response) particular solution M k=0 b k x[n k] F [n] F [n 1] F [n 2] = 0 8/10 c

Systems in TIME domain Impulse Response CT LTI Systems h(t) input CT LTI System output Convolution LTI y(t) = x(t) h(t) y(t) = x(t) h(t) = Step Response: response to u(t) x(τ)h(t τ)dτ 9/10 a

Systems in TIME domain CT LTI Systems Properties of convolution: linear, commutative, associative Simple Convolutions x(t) δ(t) = x(t) x(t) δ(t t 0 ) = x(t t 0 ) x(t) u(t) = t x(τ)dτ Graphical Approach Flip h(τ) to obtain h( τ) Shift h( τ) to right by t to obtain h(t τ) Multiple x(τ) and h(t τ) Integrate on τ to obtain x(τ)h(t τ)dτ Increase t and repeat 9/10 b

Systems in TIME domain Causal LTI Systems Described by Differential Equations Example: dy(t) dt N k=0 CT LTI Systems a k d k y(t) dt k = M k=0 b k d k x(t) dt k + 2y(t) = x(t), where x(t) = Ke 3t u(t) Important assumption for LTI and causality: initially at rest The complete solution can be written as the sum of two parts: homogeneous solution (natural response) particular solution 9/10 c

System in FREQ domain Frequency Response e jωn input DT LTI System H(e jω )e jωn output e jωt input CT LTI System H(ω)e jωt output Input-Output Relationship X(ω) H(ω) Y (ω) = X(ω)H(ω) 10/10 a

System in FREQ domain Relationship between IR and FR h(t) F H(jω) h[n] F H(e jω ) 10/10 b

System in FREQ domain Magnitude response and phase response Y (jω) = H(jω)X(jω) Y (jω) = H(jω) X(jω) Y (jω) = H(jω) + X(jω) Group delay τ(ω) = d dω { H(jω)} Bode plot for CT systems Magnitude in db, versus freq in log scale Phase in linear scale H(jω), versus freq in log scale 10/10 c

System in FREQ domain CT Ideal LPF DT FR IR inversely proportional overshoot SR ringing 10/10 d

System in FREQ domain Systems Characterized by Differential Equations N k=0 a k dk y(t) dt k = M k=0 b k dk x(t) dt k F { N k=0 a k dk y(t) dt k } = F { M } k=0 b k dk x(t) dt k N k=0 a k(jω) k Y (jω) = M k=0 b k(jω) k X(jω) H(jω) = Y (jω) X(jω) = M k=0 b k(jω) k N k=0 a k(jω) k Impulse response can be obtained by inverse FT 10/10 e

System in FREQ domain DSP of CT Signals If the sampling theorem condition is satisfied, namely, ω M < ω s 2 then the overall system is LTI, and Y c (jω) = H c (jω)x c (jω) H c (jω) = { H d (e jωt ), ω < ω s /2 0, ω > ω s /2 10/10 f