Digital Signal Processing and Analog to Digital Conversions

Johnson, K. Ch. 3 (pg. 49-58) and Slides 10 and 11 from LING 313

what is Digital Signal Processing?

• Dichotomy between continuous and discrete signals (both slightly phase-shifted)

• Time and amplitude represented in waveform

what is a Continuous Signal?

• Represented in a continuous line

• Amplitude value at all points in time

what is a Discrete Signal?

• Sequence of separate amplitude values

• Represented by separate points rather than a line

• All acoustic signals in computers are discrete

  • despite pressure fluctuations of sound are continuous

what are Analog Devices

• Instruments like tape recorders that can store a continuous signal

• They can store electric analogs of acoustic signals

• Not necessarily better at storing sounds

what are Digital Devices

Instruments that have to convert continuous speech signals into discrete signals

what are the Steps for Analog to Digital Conversion

Step 1:

• limit # of places after the decimal point on time axis

• Instead of infinite points in time, the signal is chopped into 'samples'

 

Step 2:

• limit the # of places after the decimal point on the amplitude axis

• Instead of infinitely possible levels of amplitude, signal is 'quantized'

what is the textbook definition of sampling rate?

• # of times/s we measure continuous wave in producing discrete representation of the signal

• Sample the signal often enough to capture all spectral information

  • Ex. sine wave repeats 100 times /s (100Hz)

How many samples/s do we need to take to capture the 100Hz periodic wave in a continuous signal?

at least 2 samples of each cycle of the sine wave to capture periodicity

what is the Nyquist Frequency?

• Highest-frequency component captured in a given sampling rate

• Always one-half the sampling rate (ex. sampling rate of 200Hz, the Nyquist frequency is 100Hz)

what is Aliasing?

• The misrepresentation of a continuous signal in a discrete waveform

• When continuous signal has frequency components that is higher than half the sampling rate

  • We must always filter out frequencies above half the chosen sampling rate

what is the Anti-Aliasing Filter?

low-pass filter to block frequency components in continuous signal that produces alias components

what is Quantization Noise?

• Quasi-random like white noise ( -1/5 5o 1/2 amplitude steps)

• All digital waveforms contain a degree of quantization noise

• It jumps from one quantization to the next (not continuous)

how aggressively should we round out values?

use high enough quantization rate to minimize chances of low quality recordings

how do you calculate uncompressed file size?

sampling rate x bytes per sample x seconds x channels (the higher sampling rate and bit depth can cause more storage)

what is the Signal to Noise Ratio

• Relative loudness quantization noise is lower when you use more bits to store the signal

• Ex. (8 bit sample = 256:1, 12 bit sample = 4096:1, 16 bit = 65536:1)

what is Signal Clipping?

• Bad things happen when the input volume goes beyond the range that is represented in digital waveform

• Clipping introduces transients in digital waveform

• Messes up spectral analysis you try to perform on the signal

what are the main components of sine Wave Anatomy

• amplitude A: how far the wave departs from zero

• period T: duration of one cycle, in seconds

• frequency f : cycles per second, in Hz

where does the sine wave come from?

• Right-angled triangles

• Ex. Ferris wheel

  • you constantly change position along perimeter

  • (change angle from the center)

what does it mean to say that Analog Signals are continuous

• Value exists for every time point

• Each value is precise to infinitely many decimal places

• Difficult to store/access/analyze

  • Computers only store finite list of numbers

  • Take advantage of modern computing = to digitize them

what are the 4 main steps to Analog to Digital Conversion (ADC)

• Step 1: sampling = measure signal at regular moments in time (x-axis); keep values and dump the rest

• Step 2: quantization = round each measurement to one of a limited set of values (amplitude axis)

• Step 3: after sampling have set of values

• Step 4: use discrete values stored on the computer

what is the sampling rate?

how many samples per second

if we keep 100 samples from one-second interval, out sampling rate is what?

100Hz

how do we decide which values are kept or removed?

we use the Nyquist Theorem

what is the Nyquist Theorem?

to capture the frequency f the sampling rate fs must be more than 2f (more than doubled)

What sampling rate is enough to capture frequencies up to 10 000Hz?

22, 050 (roughly) because we need just over 2f

at least how many samples need to be taken from this sine wave?

• there are 5 cycles

• 5×2 = 10, but we need a little more

• 11 or 12 samples would be the best answer!