From Information to Numbers The world is full of information Digital devices store numbers as bits How do we turn signals into numbers? Answer: Digitization –Sampling –Quantization

Sampling Sampling: Taking signal values at regularly-spaced intervals Formula: s[n] = s(nT s ) s(t) = {original signal} s[n] = {sampled signal} T s = {sampling period} f s = 1/T s is the sampling rate.

Sampling Involves a Tradeoff Too small a T s makes too many samples Too large a T s ruins the sampled signal How do we design T s ? Answer: The Nyquist Rate

The Nyquist Rate We already know: Any signal can be made from a sum of sinusoids. Suppose that a particular signal is made up of sinusoids whose frequencies are between f highest and f lowest. (f highest - f lowest ) is called the bandwidth. This signal can be exactly reconstructed from its samples if f s > 2 (f highest ). The value of 2 (f highest ) is called the Nyquist Rate.

Sampling Above and Below the Nyquist Rate f s > 2 (f highest - f lowest ) f s < 2 (f highest - f lowest ) Original Signal

Sampling Rates for Some Important Signals Designers use these sampling rates to design CD players, DVD players, cell phones, car radios, and satellite TV.

Sampling Below Nyquist Causes Aliasing Example: A 720Hz sinusoid sampled at a rate of 660Hz… …looks like a 60Hz sinusoid! Filters are used before sampling to prevent this.

Representing Text in Binary: ASCII ASCII: American Standard Code for Information Interchange A 7-bit code (8-bit representation) to represent 128 symbols –Capital and small letters (a-z, A-Z) –Numbers (0-9) –Punctuation (e.g. –Other control codes (line feed, end of file) Why would a standard code be so important?

Partial ASCII Code Listing What is ? Answer: INFINITY

Storing Samples Using Bits: Quantization Digital devices must use a limited number of bits to store each sample of a signal. The errors caused by quantization are seen and heard as noise. Example: 2 bits per sample

More Bits Mean Smaller Errors Example: –Top: 3 bits per sample –Middle: 4 bits per sample –Bottom: 16 bits per sample More bits mean higher accuracy, but more storage and effort

How Many Bits Per Sample? We would like a measure of signal quality as a function of number of bits The answer: Signal-to-Noise Ratio –Used in almost every multimedia system design Definition (linear scale) SNR = |max( )| |max( )| Definition (dB scale) SNR in dB = 20 log 10 |max( )| |max( )|

Decibel Scale Named after Alexander Graham Bell, inventor of the telephone Convenient for very large and vary small SNRs

Problem: SNR What is the SNR for the signals on the left? What is the SNR in dB? Answer: 0.8/0.06 = log 10 (80/0.6)=22.5dB

Example Problem: SNR Fact: SNR can be used to measure the quality of many signals

SNR for Quantized Signals For a B-bit quantized signal SNR = (2 B-1 )/2 -1 = 2 B Simple dB Rule: SNR = 20 log 10 (2 B ) = 6.02 B Each bit adds 6 dB to the SNR Example: CDs use 16 bits SNR CD = 96 dB