Lecture 4 Outline: Analog-to-Digital and Back Bridging the analog and digital divide Announcements: Discussion today: Monday 7-8 PM, Hewlett 102 Clarifications on PPT slides and Course Reader l PPT slides introduce the main take-away ideas from the lectures at a high level, which are repeated in more detail in the lecture l Lectures don’t perfectly follow course reader. You are only responsible for sections of the reader corresponding to lecture topics (will provide the pages before exams) Sampling vs. Analog-to-Digital Conversion (ADC) Sampling and ADC Reconstruction and Digital-to-Analog Conversion Nyquist’s Theorem

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. 0 TsTs 2T s 3T s 4T s -3T s -2T s -T s 0 TsTs 2T s 3T s 4T s -3T s -2T s -T s Today’s Focus Each level can be represented by 0s and 1s ADC DAC

x s (t) Sampling Sampling (Time): Sampling (Frequency) ADC Setting x d [n]=x s (nTs) yields X s (e j  ) with  =  T s 0 x(t) =  n  (t-nT s ) X s (j  ) X(j  ) =  n  (  -(2  n/T s )) * 0 TsTs 2T s 3T s 4T s -3T s -2T s -T s 0 TsTs 2T s 3T s 4T s -2T s -T s -3T s 2Ts2Ts -2  T s 2Ts2Ts -2  T s

Reconstruction Frequency Domain: low-pass filter Time Domain: sinc interpolation DAC LPF applied to X s (e j  ) and then converted to continuous time (  =  /T s ) recovers sampled signal X s (j  ) 0 2Ts2Ts -2  T s W -W  1  H( j  ) 0 -2  T s X s (j  ) 2Ts2Ts X r (j  ) H( j  )

Nyquist Sampling Theorem A bandlimited signal [-W,W] radians is completely described by samples every T s  /W secs. The minimum sampling rate for perfect reconstruction, called the Nyquist rate, is W/  samples/second If a bandlimited signal is sampled below its Nyquist rate, distortion (aliasing occurs) X s (j  ) X(j  ) W-W WW X(j  ) 2W=2  /T s -2W0 0

Main Points 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, ADC converts them to discrete-time signals or bits Reconstruction recreates a continuous-time signal from its samples, DAC recreates it from a discrete-time signal or bits Reconstruction in the frequency domain entails low-pass filtering; in the time-domain it entails convolution with a sinc function. A bandlimited signal of bandwidth W sampled at or above its Nyquist rate of 2W can be perfectly reconstructed from its samples

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