ESE 531: Digital Signal Processing Lec 24: April 25, 2017 Review Penn ESE 531 Spring 2017 - Khanna Course Content ! Introduction ! Oversampling, Noise Shaping ! Discrete Time Signals & Systems ! Frequency Response of LTI ! Discrete Time Fourier Systems Transform ! Basic Structures for IIR and FIR ! Z-Transform Systems ! Inverse Z-Transform ! Design of IIR and FIR Filters ! Sampling of Continuous Time ! Filter Banks Signals ! Adaptive Filters ! Frequency Domain of Discrete ! Computation of the Discrete Time Series Fourier Transform ! Downsampling/Upsampling ! Fast Fourier Transform ! Data Converters, Sigma Delta ! Compressive Sampling Modulation Penn ESE 531 Spring 2017 - Khanna 2 Digital Signal Processing ! Represent signals by a sequence of numbers " Sampling and quantization (or analog-to-digital conversion) ! Perform processing on these numbers with a digital processor " Digital signal processing ! Reconstruct analog signal from processed numbers " Reconstruction or digital-to-analog conversion digital digital signal signal analog analog signal A/D DSP D/A signal • Analog input # analog output • Eg. Digital recording music • Analog input # digital output • Eg. Touch tone phone dialing, speech to text • Digital input # analog output • Eg. Text to speech • Digital input # digital output • Eg. Compression of a file on computer Penn ESE 531 Spring 2017 - Khanna Discrete Time Signals Penn ESE 531 Spring 2017 - Khanna 4 Signals are Functions Penn ESE 531 Spring 2017 - Khanna 5 Discrete Time Systems Penn ESE 531 Spring 2017 - Khanna 6 System Properties ! Causality " y[n] only depends on x[m] for m<=n ! Linearity " Scaled sum of arbitrary inputs results in output that is a scaled sum of corresponding outputs " Ax1[n]+Bx2[n] # Ay1[n]+By2[n] ! Memoryless " y[n] depends only on x[n] ! Time Invariance " Shifted input results in shifted output " x[n-q] # y[n-q] ! BIBO Stability " A bounded input results in a bounded output (ie. max signal value exists for output if max ) Penn ESE 531 Spring 2017 - Khanna 7 LTI Systems ! LTI system can be completely characterized by its impulse response ! Then the output for an arbitrary input is a sum of weighted, delay impulse responses y[n] = x[n]∗h[n] Penn ESE 531 Spring 2017 - Khanna 8 Convolution Discrete Time Fourier Transform DTFT Definition ∞ X (e jω ) = ∑ x[k]e− jωk k=−∞ 1 π x[n] = ∫ X (e jω )e jωndω 2π −π ∞ X ( f ) = ∑ x[k]e− j2π fk k=−∞ Alternate 0.5 x[n] = ∫ X ( f )e j2π fndf −0.5 Penn ESE 531 Spring 2017 - Khanna 11 Example: Window DTFT ∞ W (e jω ) = ∑ w[k]e− jωk k=−∞ N = ∑ e− jωk k=−N Penn ESE 531 Spring 2017 - Khanna 12 Example: Window DTFT sin (N +1 2)ω W (e jω ) = ( ) Also, Σx[n] sin(ω 2) =1 why? Plot for N=2 Penn ESE 531 Spring 2017 - Khanna 13 LTI System Frequency Response ! Fourier Transform of impulse response x[n]=ejωn LTI System y[n]=H(ejωn)ejωn ∞ H(e jω ) = ∑ h[k]e− jωk k=−∞ Penn ESE 531 Spring 2017 - Khanna 14 z-Transform ! The z-transform generalizes the Discrete-Time Fourier Transform (DTFT) for analyzing infinite- length signals and systems ! Very useful for designing and analyzing signal processing systems ! Properties are very similar to the DTFT with a few caveats Penn ESE 531 Spring 2017 - Khanna 15 Complex Exponentials as Eigenfunctions zn H(z)zn H(z)zn Penn ESE 531 Spring 2017 - Khanna 16 Region of Convergence (ROC) Penn ESE 531 Spring 2017 - Khanna 17 Formal Properties of the ROC ! PROPERTY 1: " The ROC will either be of the form 0 < rR < |z|, or |z| < rL < ∞, or, in general the annulus, i.e., 0 < rR < |z| < rL< ∞. ! PROPERTY 2: " The Fourier transform of x[n] converges absolutely if and only if the ROC of the z-transform of x[n] includes the unit circle. ! PROPERTY 3: " The ROC cannot contain any poles. ! PROPERTY 4: " If x[n] is a finite-duration sequence, i.e., a sequence that is zero except in a finite interval -∞ < N1 < n < N2 < ∞, then the ROC is the entire z- plane, except possibly z = 0 or z = ∞. Penn ESE 531 Spring 2017 - Khanna 18 Formal Properties of the ROC ! PROPERTY 5: " If x[n] is a right-sided sequence, the ROC extends outward from the outermost finite pole in X(z) to (and possibly including) z = ∞. ! PROPERTY 6: " If x[n] is a left-sided sequence, the ROC extends inward from the innermost nonzero pole in X(z) to (and possibly including) z=0. ! PROPERTY 7: " A two-sided sequence is an infinite-duration sequence that is neither right sided nor left sided. If x[n] is a two-sided sequence, the ROC will consist of a ring in the z-plane, bounded on the interior and exterior by a pole and, consistent with Property 3, not containing any poles. ! PROPERTY 8: " The ROC must be a connected region. Penn ESE 531 Spring 2017 - Khanna 19 Inverse z-Transform ! Ways to avoid it: " Inspection (known transforms) " Properties of the z-transform " Power series expansion " Partial fraction expansion Penn ESE 531 Spring 2017 - Khanna 20 Partial Fraction Expansion ! Let M M b z−k z N b zM −k ∑ k ∑ k X (z) k=0 k=0 = N = N a z−k zM a z N−k ∑ k ∑ k k=0 k=0 ! M zeros and N poles at nonzero locations Penn ESE 531 Spring 2017 - Khanna 21 Partial Fraction Expansion ! If M<N and the poles are 1st order M (1− c z−1) b ∏ k N A X (z) = 0 k=1 = k N ∑ −1 a 1− d z 0 (1 d z−1) k=1 k ∏ − k k=1 ! where −1 Ak = (1− dk z )X (z) z=dk Penn ESE 531 Spring 2017 - Khanna 22 Example: 2nd-Order z-Transform ! 2nd-order = two poles Right sided −1 2 ⎧ 1 ⎫ X (z) = + , ROC = ⎨z : < z ⎬ ⎛ 1 −1⎞ ⎛ 1 −1⎞ ⎩ 2 ⎭ ⎜1− z ⎟ ⎜1− z ⎟ ⎝ 4 ⎠ ⎝ 2 ⎠ n n ⎛ 1⎞ ⎛ 1⎞ x[n] = −⎜ ⎟ u[n]+ 2⎜ ⎟ u[n] ⎝ 4⎠ ⎝ 2⎠ Penn ESE 531 Spring 2017 - Khanna 23 Partial Fraction Expansion ! If M≥N and the poles are 1st order M −N N A X (z) = B z−r + k ∑ r ∑ −1 r=0 k=1 1− dk z ! Where Bk is found by long division −1 Ak = (1− dk z )X (z) z=dk Penn ESE 531 Spring 2017 - Khanna 24 Example: Partial Fractions ! M=N=2 and poles are first order A A X (z) = B + 1 + 2 , ROC = z : 1< z 0 −1 { } 1 1 1 z 1− z− − 2 2 1 3 z−2 − z−1 +1 z−2 + 2z−1 +1 2 2 z−2 − 3z−1 + 2 −1 1 5z −1 −1+5z− X (z) = 2 + 1 (1− z−1)(1− z−1) 2 Penn ESE 531 Spring 2017 - Khanna 25 Example: Partial Fractions ! M=N=2 and poles are first order 9 8 X (z) = 2 − + , ROC = z : 1< z −1 { } 1 1 1 z 1− z− − 2 n ⎛ 1⎞ x[n] = 2δ[n]− 9⎜ ⎟ u[n]+8u[n] ⎝ 2⎠ Penn ESE 531 Spring 2017 - Khanna 26 Power Series Expansion ! Expansion of the z-transform definition ∞ X (z) = ∑ x[n]z−n n=−∞ =!+ x[−2]z2 + x[−1]z + x[0]+ x[1]z−1 + x[2]z−2 +! Penn ESE 531 Spring 2017 - Khanna 27 Example: Finite-Length Sequence ! Poles and zeros? 2 ⎛ 1 1⎞ 1 1 X (z) = z ⎜1− z− ⎟(1+ z− )(1− z− ) ⎝ 2 ⎠ ⎧ 1, n = −2 2 1 1 −1 ⎪ = z − z −1+ z ⎪ −1 2, n = −1 2 2 ⎪ x[n] = ⎨ −1, n = 0 ∞ ⎪ 1 2, n =1 −n ⎪ X (z) = x[n]z ⎪ 0, else ∑ ⎩ n=−∞ =!+ x[−2]z2 + x[−1]z + x[0]+ x[1]z−1 + x[2]z−2 +! Penn ESE 531 Spring 2017 - Khanna 28 Example: Finite-Length Sequence ! Poles and zeros? 2 ⎛ 1 1⎞ 1 1 X (z) = z ⎜1− z− ⎟(1+ z− )(1− z− ) ⎝ 2 ⎠ 1 1 = z2 − z −1+ z−1 2 2 ⎧ 1, n = −2 ⎪ ⎪ −1 2, n = −1 ⎪ 1 1 x[n] = ⎨ −1, n = 0 = δ[n + 2]− δ[n +1]−δ[n]+ δ[n −1] 2 2 ⎪ 1 2, n =1 ⎪ ⎪ 0, else ⎩ Penn ESE 531 Spring 2017 - Khanna 29 Difference Equation to z-Transform N M a y[n k] b x[n m] ∑ k − = ∑ m − k=0 m=0 N ⎛ ⎞ M ⎛ ⎞ ak bk y[n] = −∑⎜ ⎟ y[n − k]+∑⎜ ⎟x[n − m] k=1⎝ a0 ⎠ k=0⎝ a0 ⎠ ! Difference equations of this form behave as causal LTI systems " when the input is zero prior to n=0 " Initial rest equations are imposed prior to the time when input becomes nonzero " i.e y[-N]=y[-N+1]=…=y[-1]=0 Penn ESE 531 Spring 2017 - Khanna 30 Difference Equation to z-Transform N ⎛ ⎞ M ⎛ ⎞ ak bk y[n] = −∑⎜ ⎟ y[n − k]+∑⎜ ⎟x[n − m] k=1⎝ a0 ⎠ k=0⎝ a0 ⎠ N ⎛ ⎞ M ⎛ ⎞ ak −k bk −k Y (z) = −∑⎜ ⎟z Y (z) +∑⎜ ⎟z X (z) k=1⎝ a0 ⎠ k=0⎝ a0 ⎠ M b z−k N ⎛ a ⎞ M ⎛ b ⎞ ∑( k ) ⎜ k ⎟z−kY (z) = ⎜ k ⎟z−k X (z) ⇒ Y (z) = m=0 X (z) ∑⎜ a ⎟ ∑⎜ a ⎟ N k=0⎝ 0 ⎠ k=0⎝ 0 ⎠ a z−k ∑( k ) M k=0 b z−k ∑( k ) H(z) m=0 = N a z−k ∑( k ) Penn ESE 531 Spring 2017 - Khanna k=0 31 st M Example: 1 -Order System b z−k ∑( k ) H(z) m=0 = N a z−k ∑( k ) k=0 y[n] = ay[n −1]+ x[n] 1 H(z) = h[n] = anu[n] 1− az−1 Penn ESE 531 Spring 2017 - Khanna 32 Sampling Penn ESE 531 Spring 2017 - Khanna 33 DSP System Penn ESE 531 Spring 2017 - Khanna 34 Ideal Sampling Model ! Ideal continuous-to-discrete time (C/D) converter " T is the sampling period " fs=1/T is the sampling frequency " Ωs=2π/T Penn ESE 531 Spring 2017 - Khanna 35 Ideal Sampling Model Penn ESE 531 Spring 2017 - Khanna 36 Frequency Domain Analysis jΩ X s (e ) ω = ΩT Ω s ⋅T = π 2 Penn ESE 531 Spring 2017 - Khanna 37 Frequency Domain Analysis jΩ X s (e ) ω = ΩT Ω s ⋅T = π 2 Penn ESE 531 Spring 2017 - Khanna 38 Aliasing Example Penn ESE 531 Spring 2017 - Khanna 39 Reconstruction of Bandlimited Signals ! Nyquist Sampling Theorem: Suppose xc(t) is bandlimited.