Lecture 5: Signal Processing II EEN 112: Introduction to Electrical and Computer Engineering Professor Eric Rozier, 2/20/13

SOME DEFINITIONS

Decibels Logarithmic unit that indicates the ratio of a physical quantity relative to a specified level. 10x change is 10 dB change. 2x change ~3dB change. – Remember L_dB = 10 log_10 (P1/P0) for power L_db = 20 log_10 (A1/A0) for amplitude (Power ~ Amplitude^2)

Period A measurement of a time interval A periodic signal that repeats every 10s Periodic observation, count the number of students who are asleep every 1 minute

Rate 1/period If I count the number of students who are asleep every minute, I do so with the rate of 1/60s, or at a rate of Hertz

Hertz Instances per second kHz, MHz, GHz – standard SI-prefixes for hertz

Rate and Time If a period is 10s, the rates is 1/10s. Hertz is cycles per second

Bandwidth (signal processing) Difference between the upper and lower frequencies in a continuous set that carry information of interest. Not to be confused with data bandwidth, which while related is not the same concept

SAMPLING CONTINUOUS SIGNALS

Sampling Conversion of continuous time signals into discrete time signals. How frequently we record, witness, or store, some signal. Frame rates, movies typically play at 24 frames/second (rate) – What is the period?

Sampling Affects how much data we have to store to represent a signal. The more we store, the more space it takes! The less we store, the more error is introduced! How do we know how much is enough?

Digital Sampling

Sampling Issues

The Problem

Fixing the Problem

Sampling Nyquist Theorem (sampling theorem) – An analog signal of bandwidth B Hertz when sampled at least as often as once every 1/2B seconds (or at 2B Hertz), can be exactly converted back to the analog original signal without any distortion or loss of information. This rate is called the Nyquist sampling rate.

Nyquist in Practice Telephone speech has a bandwidth of 3500 Hz – At what rate should it be sampled? – 7000 Hz – In practice it is sampled at 8000 Hz, to avoid conversion factors (Once every 124 microseconds)

Acoustic Signals Acoustic signals are audible up to 24 kHz – What is the corresponding Nyquist sampling rate?

Acoustic Signals Industrial standards – 6000 Hz – 8000 Hz – Hz – Hz – Hz – Hz – Hz – Hz – Hz

Spoken Sentence

16000 Hz Hz 8000 Hz 6000 Hz

Piano

Spoken Sentence Hz Hz 8000 Hz 6000 Hz

SPECTROGRAMS

Spectrogram Visual representation of frequencies in a signal. Sometimes called, spectral waterfalls, or voiceprints/voicegrams Can identify spoken words phonetically. Also used in sonor, radar, seismology, etc.

Spectrogram Frequency vs Time Color or height mapped to dB

Spectrogram – Speech Hz

Spectrogram – Speech Hz

Spectrogram – Speech 8000 Hz

Spectrogram – Speech 6000 Hz

Spectrogram – Piano Hz

Spectrogram – Piano Hz

Spectrogram – Piano 8000 Hz

Spectrogram – Piano 6000 Hz

ANALOG TO DIGITAL CONVERSION

A2D: Analog to Digital Two steps – Sampling (which we just covered) – Quantization

Quantization Analog signals take any value between some minimum and maximum – Infinite possible values – We need a finite set of values

Why do we need finite values?

State in Digital Logic Flip-flops store state for sequential logic (vs combinatorical logic) Each one can hold a 0 or 1, one bit Put X together and we have X bits worth of state we can store

How do we get this?

How to quantize Informally – If we have N bits per value, we have how many states? – Values from [min, max] (inclusive) – Each state provided by our bit vector needs to cover of the range

How to quantize Simple algorithm, assume 2-bits, how many states?

How to quantize Simple algorithm, assume 2-bits, how many states? First state is min. We now have (4-1) = 3 states left to cover the range (Max – Min) 00 – Min 01 – Min + (Max – Min)/3 10 – Min + 2(Max – Min)/3 11 – Min + 3(Max – Min)/3 = Max

How to quantize What do we do with data in between these values? Let’s refine our algorithm

Quantization Classification rule – Tells us which state of our bit vector the value corresponds to Reconstruction rule – Tells us how to interpret a state of the bit vector

Quantization Classification Rule A general classification rule

Quantization Reconstruction Rule A general reconstruction rule

Putting it all Together From 5 to 12, 2-bits

Homework See course website for this week’s signals homework.