Lecture 2 Outline: Analog-to-Digital and Back Bridging the analog and digital divide Announcements: Poll for discussion section and OHs: please respond First HW posted tonight Lectures going forward: PPT slides and reader material PPT slides introduce main take-away ideas from lectures at a high level, which are repeated in more detail in the lecture Lectures don’t perfectly follow course reader. You are responsible for lecture material and reader sections covered in lecture (will provide pages before exams) Sampling vs. Analog-to-Digital Conversion (ADC) Sampling and ADC Reconstruction and Digital-to-Analog Conversion Nyquist Sampling Theorem Quantization
Review of Last Lecture Course Overview EE102a Review Signals and Systems in the Time Domain Continuous, Discrete, and Hybrid Continuous-Time Signals in the Frequency Domain Fourier Series and Fourier Transforms Discrete Time Signals in the Frequency Domain Duality Relationships Connections between Continuous/Discrete Time Filtering and Convolution Periodic Signals Energy, Power, and Parseval
Sampling and Reconstruction vs Sampling and Reconstruction vs. Analog-to-Digital and Digital-to-Analog Conversion Sampling: converts a continuous-time signal to a continuous-time 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)p(t) X(jw)*P(jw)/(2p) Analog-to-Digital Conversation (ADC) Setting xd[n]=xs(nTs) yields Xs(ejW) with W=wTs x(t) nd(t-nTs) = xs(t) Ts 2Ts 3Ts 4Ts -3Ts -2Ts -Ts -3Ts -2Ts -Ts Ts 2Ts 3Ts 4Ts X(jw) * = nd(w-(2pn/Ts)) Xs(jw) -2p Ts 2p Ts -2p Ts 2p Ts
Reconstruction Frequency Domain: low-pass filter Time Domain: sinc interpolation Digital-to-Analog Conversation (DAC) LPF applied to Xs(ejW) and then converted to continuous time (w=W/Ts) recovers sampled signal H(jw) Xs(jw) -2p Ts Xs(jw) 2p Xr(jw) H(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 Tsp/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
Quantization x(t) xQ(nTs) A … -A+kD … -A+2D -A+D -A Ts 2Ts Divide amplitude range [-A,A] into 2N levels, {-A+kD}, k=0,…2N-1 Map x(t) amplitude at each Ts to closest level, yields xQ(nTs)=xQ[n] Convert k to its binary representation (N bits); converts xQ[n] to bits
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 Multiplication with delta train in time, convolution with delta train in frequency ADC converts a continuous-time signal to a discrete-time signal or bits Reconstruction recreates a continuous-time signal from its samples Multiplication with LPF in frequency, sinc interpolation in time DAC recreates a continuous-time signal 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 Quantization converts a discrete-time signal to bits by mapping its values to a finite number of levels, which introduces noise