Lecture 3 Outline: Sampling and Reconstruction Announcements: Discussion: Monday 7-8 PM, Location (likely) Hewlett 102 TA OHs (will be held in kitchen area outside Packard 340): Alon 6-7 PM on Monday Mainak 4-5 PM on Tuesday, 1-2 PM on Wednesday Jeremy 3-4 PM on Friday, 3-4 PM on Tuesday. First HW posted, due next Wed. 5pm TGIF treats Finish “Fun with Fourier” Sampling Reconstruction Nyquist Sampling Theorem

Review of Last Lecture Duality Relationships: Connections between Continuous/Discrete Time Discrete time less intuitive than continuous time (for most people) Can consider a discrete time process as samples of a continuous time process, which is periodic in frequency Filtering (clarification to Case 2: W=2p/T=6p/T0) .5T -.5T A t x(t) AT w X(jw) AT t x(t) .5W -.5W AW w=2pf X(jw) Mathematically Sound Definition LPF: X(±jW)=.5 Similarly for P(t) a2 a1 a0 a3 w a4 a-2 a-3 a-4 a5 a6 a-5 a-6 a-1 Include a3,a-3? T0 -T0 .5T -.5T … X(jw) Depends on value of X(jw) at w=W 1 0.5 If X(±jW)>0, yes, if zero then no Assumed last lecture -W W W

Filtering and Convolution Filtering Example: Discrete Periodic Pulse Important convolution examples OWN pg. 392 2W a0 x[n]: Periodic Discrete X(ejW) a-1 a1 a-4 W -W 1 a4 -N -N1 N1 N n a-3 a3 a-2 a2 NA2 3A2 3A2 2T A2T 2A2 2A2 T A T A A * = * = A2 A2 … … … … N-1 N-1 N-1 2N-1

Periodic Signals Continuous time: Discrete time: … … … … x(t) periodic iff there exists a T0>0 such that x(t)=x(t+T0) for all t T0 is called a period of x(t); smallest such T0 is fundamental period If x(t) periodic with period T0, y(t) periodic with period T1, then x(t)+y(t) is periodic with period k0T0=k1T1 if such integers ki exist. Discrete time: x[n] periodic iff there exists a N0>0 such that x[n]=x[n+N0] for all n N0 is called a period of x(t); smallest such N0 is fundamental period If x[n] periodic with period N1, y[n] periodic with period N2, then x[n]+y[n] is periodic with period k1N1=k2N2 if such integers ki exist. T0 2T0 -T0 .5T -.5T t k0T0 T0 … 2T0 k1T1 T1 … 2T1 -N0 -N N N0 n … k1N1 N1 … 2N1 2N0 k0N0 N0

Energy, Power, and Parseval’s Relation Energy signals have zero power; Power signals have infinite energy Continuous Time Aperiodic Signals Discrete Time Aperiodic Signals −∞ ∞ |𝑥 𝑡 | 2 𝑑𝑡= 1 2𝜋 −∞ ∞ |𝑋 𝑗𝜔 | 2 𝑑𝜔 Periodic: 𝑛=−∞ ∞ |𝑥 𝑛 | 2 = 1 2𝜋 2𝜋 |𝑋 𝑒 𝑗 | 2 𝑑 Periodic:

Examples Continuous-time Discrete-time x(t) X(jw) AT w Which would you rather integrate? a0 a1 a2 a3 a4 a-2 a-1 a-3 a-4 N 2N -N N1 -N1 n x[n]: Periodic Discrete Which would you rather sum?

Sampling and Reconstruction vs Sampling and Reconstruction vs. Analog-to-Digital and Digital-to-Analog Conversion Sampling: converts a continuous-time signal to a sampled signal Reconstruction: converts a sampled signal to a continuous-time signal. Analog-to-digital conversion: converts a continuous-time signal to a discrete-time quantized or unquantized signal Digital-to-analog conversion. Converts a discrete-time quantized or unquantized signal to a continuous-time signal. Ts 2Ts 3Ts 4Ts -3Ts -2Ts -Ts Ts 2Ts 3Ts 4Ts -3Ts -2Ts -Ts 1 2 3 4 -3 -2 -1 Each level can be represented by 0s and 1s 1 2 3 4 -3 -2 -1

Applications Capture: audio, images, video Storage: CD, DVD, Blu-Ray, MP3, JPEG, MPEG Signal processing: compression, enhancement and synthesis of audio, images, video Communication: optical fiber, cell phones, wireless local-area networks (WiFi), Bluetooth Applications: VoIP, streaming music and video, control systems, Fitbit, Occulus Rift

Sampling Sampling (Time): = Sampling (Frequency) = * x(t) nd(t-nTs) xs(t) Ts 2Ts 3Ts 4Ts -3Ts -2Ts -Ts -3Ts -2Ts -Ts Ts 2Ts 3Ts 4Ts = X(jw) * nd(t-n/Ts) Xs(jw) -2p Ts 2p Ts -2p Ts 2p Ts

Reconstruction Frequency Domain: low-pass filter Time Domain: sinc interpolation H(jw) Xs(jw) H(jw) -2p Ts Xs(jw) 2p Xr(jw) 1 -W W w w -2p Ts 2p Ts

Nyquist Sampling Theorem A bandlimited signal [-W,W] radians is completely described by samples every Tsp/W secs. The minimum sampling rate for perfect reconstruction, called the Nyquist rate, is W/p samples/second If a bandlimited signal is sampled below its Nyquist rate, distortion (aliasing occurs) X(jw) X(jw) Xs(jw) -W W -2W W W 2W=2p/Ts

Main Points Sum of periodic signals is periodic if integer multiples of each period are equal Energy and power can be computed in time or frequency domain Sampling bridges the analog and digital worlds, with widespread applications in the capture, storage, and processing of signals Sampling converts continuous-time signals to sampled signals Multiplication with delta train in time, convolution with delta train in frequency Reconstruction recreates a continuous-time signal from its samples Multiplication with LPF in frequency, sinc interpolation in time A bandlimited signal of bandwidth W sampled at or above its Nyquist rate of 2W can be perfectly reconstructed from its samples