1	Introduction to Digital Sampling and Signals Timothy Hoerning Spring 2006
2	Overview Background math Visual aids Analog Signals and Recording Digital Signals Sampling into discrete time Quantizing Bits and bytes – PCM encoding Error and Overflow Digital Filtering Digital Effects
3	Bases A Base determines the maximum number represented by any digit in that system Base 10 uses the digits 0—9 Base 2 uses the digits 0—1 Base 8 uses the digits 0—7 Base 16 uses the digits 0—9 and A—F Base 10 are the decimal digits we are all familiar with. Base 2 is used in the binary number system represented by computers Bases 8 and 16 are used as short hand for representing binary numbers
4	Powers & Logarithms Powers – Multiply by itself times Logarithms – the base logarithm of is (the reverse of powers)
5	Bits and Bytes We as humans have evolved a system of decimal digitals (due in part to having 10 fingers or digits) A digital computer operates with just two states for any digit. Hence it uses binary digitals A bit is shortening of a binary digit. A byte is a short for of of “by eight”. It is a group of 8 bits.
6	Number System Comparison Example: Decimal Each column in the number represents a different power of 10 1237 = 1 * 10³ = 11000 + 2 10² = 2100 + 3 10¹ = 310 + 7 10⁰ = 71 Example: Binary Each column in the number represents a different power of 2 10011010101 = 1 2¹⁰= 11024 0 2⁹ = 0512 0 2⁸ = 0256 1 2⁷ = 1128 1 2⁶= 164 0 2⁵ = 032 1 2⁴= 116 0 2³= 08 1 2²= 14 0 2¹= 02 1 2⁰= 1*2
7	A word about Overflow All binary systems have a fixed number of bits available to them If a calculation exceeds these number of bits, an overflow condition has arisen. Overflows are typically flagged within the CPU, but at that point it is too late as the data causing the overflow has already been combined. Overflows will be useful in 2’s complement math as we shall see. Digital Signal Processors use accumulator registers with extra precision to combat overflow when multiple Multiply and ACcumulate (MAC) are used in succession.
8	decibels & SNR Given the wide range of acceptable sounds, it is often beneficial to represent the ratio of two signals in decibels If the upper signal is the desired signal voltage and the lower signal is the noise voltage, than the ratio becomes a signal to noise (SNR) ratio
9	Time and Frequency Responses Two common method are used for visualizing signals Time Domain Frequency Domain MATLAB’s xpsound is a good introduction to visualization techniques
10	Spectrogram = time & frequency MATLAB’s xpsound demo allows the user to explore the time domain, freq domain and both at once spectrogram The signal shown on the bottom left is the dropping egg. The curve in the left hand section represents the falling pitch as the egg drops The line about 2/3 of the way through is the “splat” impulse. Notice the broadband nature of the impulse
11	What is Analog? Sound is the compression and rarefaction of air molecules An analog signal is an electrical signal where the voltage is analogous to the input waveform
12	What is it about Analog? Analog was the only form of signal processing for many years. Several methods of analog recording have come and passed Wax cylinders and disks Vinyl Records Magnetic Tape 8 tracks Cassettes Analog typically involves some type of distortion. This is often considered as “warmth” by analog purists RIAA preamp for records Saturation of magnetic tape The main problem is that analog is noisy (and distorted, etc). A certain amount of this noise and distortion is added with each additional generation. Digital does not have this problem.
13	What is Digital? Digital is concept; it is an abstraction. There is no such thing as a “digital signal” versus an “analog signal” A digital signal is assumed to have one of N levels The simplest signal uses 2 levels. This is called a binary system and is what we will limit ourselves to in this presentation. All data is transmitted or stored as a analog representation Cables carry voltages or currents Hard drives store things using magnetic particles Computer Memory stores electric charge Error free digital requires that the analog signal is not so degraded in the process that it cannot be received uncorrupted.
14	Why Digital? Resistance to noise As long as no errors are introduced, signal can be reconstructed and duplicated perfectly If error correction is used, errors can be removed. The correct data can be recovered in the presense of noise.
15	How do we make a signal digital? An analog signal exists in continuous time, continuous amplitude and the frequency can potentially vary from –infinity to infinity To make a signal digital, we must to two things Sample the signal into Discrete Time Quantize the levels
16	Discrete time Signals can be converted into discrete time through the use of a sample and hold device. This device samples the signal at a sampling rate (fs) which is much greater than the input sampling rate of the input waveform. Each sampled signal is held until the next sample is taken This sampling can be used with “bucket brigade” (discrete time) devices or as an input to a quantizer (to be converted to a digital signal)
17	Nyquist Sampling Theory Nyquist Sampling Theory states that the sampling rate of a system must be twice the maximum input frequency fs=2*BW In practice, because of realistic filtering, the maximum frequency is limited to less than1/2 If phase detection is required, then an even higher sampling rate should be used. Standard sampling frequencies 8kHz – Telephone network – speech signals are nominally in the 300Hz to 3.4kHz range 44,100Hz – CD quality audio – theoretically 22kHz signals, ~16kHz in reality 98kHz – new emerging standard for super high fidelity recording
18	Aliasing & Imaging If a signal that exceeds the Nyquist sampling rate, it will be incorrectly “aliased” to the wrong frequency in the digital domain. This is not common on commercially available audio equipment since they include front end filters. For this reason, the digital domain is considered to have a periodic frequency domain. The valid range is from –fs/2 to fs/2. Outside this are images. If a signal manipulation in the digital realm causes frequencies to be adjusted or shifted, they will also be aliased to other frequencies When the signal is created via a D/A, it must be filtered to reduce the out of band energy. These out of band signals are called “images”.
19	PCM The most common form for storing a digital signal (and the only one we will consider) is Pulse Code Modulation or PCM Simply, PCM assigns a certain code to each sample value 0 = 000 1 = 001, etc. Negative numbers are expressed in 2’s complement.
20	2’s complement There are multiple methods of expressing negative numbers in binary. One of the more common is 2’s complement. This is the only one we will consider in this presentation Another method, sign magnitude, is common in digital signal processors. Consider a k bit system Positive numbers 0—2^(k-1)are simply their binary equivalent Negative numbers are -2^k— -1are set to the binary number that will yield zero when added to the positive version. (ex, in a 3 bit system, 3 = 011, then -3 would need to be 101 because 011 + 101 = 1000, but since it is three bits, there is an overflow and that is really 000 or zero)
21	2’s complement chart Consider a 4 bit system as shown in the graph on the left Note that the most significant bit is only one for the negative numbers Note that the system has one more negative number than positive. This number is typically avoided
22	Quantization Quantizing a signal makes it discrete amplitude. Each held voltage is measured and the closest level is selected as the level for that sample A k bit symbol takes on N=2^k possible values. (ex, 8 bit signals have 256 levels and 16bit signals have 65,535 levels) The example shown below is a 3 bit system
23	Quantization Noise The rounding process adds what is referred to as quantization noise. Another way to think about the quantization process is to consider the input discrete time signal as your desired signal. Consider an imaginary signal that is when added to the D.T. signal will create the digital signal. This imaginary signal is the noise signal.
24	Quantization Noise (cont) The higher the number of quantization levels, the lower the quantization noise. For a low number of levels (~<6 bits), the quantization noise can be heard as correlated with the input signal For larger number of bits, the quantization noise can be modeled as an independent Gaussian noise source. maxSNR=20*log₁₀(2^k) (ex, 16bit sound has a dynamic range (or best case SNR of 96dB)
25	What does Quantization Sound like?
26	Overflow & Headroom If a signal exceeds the maximum value represented by the k-bit system, then an overflow has occurred. If this is detected, the signal can be clipped to the maximum level. This will result is some distortion, but depending on the level of the clip it may be tolerable If unchecked, the signal will wrap around from the positive side of the y-axis to the negative because of the 2’s complement implementation. This will cause a noticeable pop in the output waveform. The amount of signal level between the normal operating point and the clipping point (or the input overflow point) is the headroom. Care should be taken in selecting an input power level. Too low and the quantization noise will be apparent Too high and the headroom will be too low to handle the dynamics of the signal
27	Errors & Error Correction The read state of a digital signal is determined by a threshold Above the threshold and the value is read as a 1 Below the threshold and the value is read as a 0 Once a bit is read incorrectly, it cannot be corrected unless error correction is used Actual digital devices usually don’t use a simple threshold. Instead they have a valid 1 region and a valid 0 region with a “dead zone” between them. Error Correction Used on some common formats such as CDs Adds overhead. Required storage can be more than 2x that required to store data unprotected Still has it’s limits. Nothing can correct all errors.
28	Not all bits are created equal Remember that with PCM, the Most significant bit determines whether a signal is above or below the x axis. An error on this bit will create a large error The Least Significant Bit however determines which of two adjacent levels is selected An error on this bit will create a barely noticeable error Sometimes Errors are intentionally introduced on the least significant bits to encode extra information Robbed Bit Signaling – used the Public Switched Telephone Network (PSTN) to indicate the status of the switch hook. The least significant bit is “robbed” on every 1 out of 6 frames to encoded this. Audio Watermarking – used as a form of copy protection on CDs.
29	Digital Effects
30	Digital Effects Effects are used for one of two things: to imitate a sound but in a smaller form factor To create a new sound not previously realizable without the effect Many effects can be realized in the Analog or the digital domain The sound differences usually arise from “imperfections” in the analog implementation Many people are used to the imperfections and find them more pleasing Some effects are uniquely digital such as delay Sometimes discrete time effects (such as bucket brigade based delay pedals) are described as “analog”. This is a bit of a misnomer since the signals are sampled. These effects to suffer from distortion on very long delay times
31	Tremolo Multiply the input signal by the output of a Low Frequency Oscillator (LFO). (maybe analog or digital) LFO can vary in shape (sine, square, etc.), frequency, amplitude (typically referred to as depth) Can also use different phase LFOs to get a stereo effect. Graphs at right show Tremolo with sin and triangle wave modulation. Depth is set to about 50% and the modulating signal is at 2Hz.
32	Ring Modulator Same notional schematic as the Tremolo. Instead of LFO with variable depth, use audio freq with full amplitude. (may be analog or digital) Output includes the sum and difference frequencies. Very non-musical effects. Sum and difference frequencies are not musically related to input. Spectrogram below shows a sine-wave major scale starting at A=220Hz and the output when mixed with a 400Hz fixed frequency.
33	Filtering Basics Filters are designed to allow some frequencies to pass unaltered (the pass band), while some are attenuated (the stop band). Filters are described by the size and location of the bands and the attenuation possible in the stop band
34	Digital Filtering Digital Filters can be implemented in one of two ways: feed forward only (Finite Impulse Response - FIR) and feed forward and feed back (Infinite Impulse Response - IIR) FIR filters are usually used because they are easier to design. IIR designs can lead to problems with instability. MATLAB’s filtdemo provides an interactive method of learning about FIR and IIR
35	Active Filters Active Filters by definition require a power source. Passive filters can only remove frequency components. Active filters can boost as well as cut frequency bands. Also allow for sharper transitions Allow multi-pole filters using solid state devices and capacitors instead of inductors (allows for implementation in integrated circuits)
36	Basic Parameters Break Frequencies (or center frequency) – The frequencies at which the response of the filter changes slope. (i.e. the poles and zeros) Bandwidth – The width of the filter in the frequency domain. Usually defined at the point 3dB below the peak signal amplitude. Q – The ratio of the center frequency to the bandwidth. Q is generally fairly low for audio applications Boost/Cut Amplitude – Amount of amplification or attenuation obtained by a filter. Typically specified in dB with a range of +/- 10dB to +/- 20dB
37	Equalization Named because it was designed to “equalize” the response of the room. The goal was to use the notches and peaks of the equalizer to compensate for resonances and dead spots within a room Types Graphic Fixed center frequency, fixed Q, variable amplitude Spaced in octaves (or fraction of octaves, 1/2 octave or 1/3 octave) Parametric Variable center frequency, variable Q, variable amplitude Typically only 3 or 4 bands Semi-Parametric Variable center frequency, fixed Q, variable amplitude
38	Parametric Equalization Response
39	“Analog” vs. Digital Many guitarists talk about the difference between “analog” versus digital delay based devices In truth, there are no completely analog delay circuits (except for tape-echo units). In “analog” delay pedals, the delay unit is a bucket-brigade chip A bucket-brigade chip is an integrated circuit with a large number of capacitors and transmission gate transistors. The signal is sampled (into discrete time), and the sampled voltage is placed on the first capacitor The sampled voltage is then passed to the next capacitor at the rate determined by the sampling rate control. If the sampling rate is too low, the voltage on the capacitor will “bleed” down and distortion will occur. This gives the signal a “warm” quality that many appreciate.
40	Delay
41	More Delay
42	Echo Often there is no distinction between delay units and echoes. What usually makes a device more of an echo is how it attempts to simulate a tape echo The feedback gain can be increased to be greater than unity. This will cause the echoes to increase in volume and eventually saturate or oscillate The is some times a filter in the feedback loop to simulate the gradual deterioration of signal quality that would occur with a tape-echo unit. Sometimes the delay time is varied slightly to simulate tape warble.
43	Frequency Response While it’s not usually noticeable, simple delays are frequency selective. It should be obvious that a frequency and all its odd harmonics will destructively interfere and cause nulls in the frequency domain
44	Computer Implementation of Delay Delays in a computer (or Digital Signal Processor) are implemented using a circular buffer An input pointer writes the sampled values into memory An output pointer reads the sample values out of memory a fixed number of samples later. The offset in samples times 1/fs is the delay time DSPs have special registers that automatically wrap around the end of a circular buffer.
45	Reverb Reverb is designed to sound like the multiple reflections found in a room or a hall. Many analog reverbs were created using a spring and transducers “True-convolution” is the newest trend in reverbs. This uses an impulse response from a real location. The reverb is effected by using a very long FIR filter Can use an FIR filter for early reflections with an IIR filter or the trailing reflections
46	Flanger As was noticed previously, delay elements can create notches in the frequency domain. By changing the delay time continuously (at a sub-sampled value), the notches vary in frequency The delay time is modulated with a low frequency oscillator (LFO). The following parameters determine the sound from a flanger Delay - The minimum amount of time the signal is delayed. (100 ms to 10 ms). Sweep depth (width) - How wide the sweep is in terms of delay time. i.e., the maximum additional delay time added as the LFO is swept (100 ns to 10 ms) Speed – the frequency of the LFO
47	Flanger Response
48	Chorus Named because it makes one instrument sound like many. Fills up the sonic space in a song A chorus is merely a flanger with longer delay times (and usually no feedback) The following parameters determine the sound from a flanger Delay - The minimum amount of time the signal is delayed. (20-30 ms). Sweep depth (width) - How wide the sweep is in terms of delay time. i.e. the maximum additional delay time added as the LFO is swept (0 to 10 ms) Speed – the frequency of the LFO Extreme settings will create a vibrato – pitch warble
49	Chorus Responses
50	Music & Engineering ECE313 Course for Junior EEs (or those with appropriate signal processing and circuits background) Topics Covered Music Theory Reading the staff, building scales and chords Just and Equal temperament – sine waves to notes Effects processing Simulation of common effects in MATLAB (distortion, chorus, flanging, delay, ring modulation, etc) Audio Synthesis MIDI (Musical Instrument Digital Interface Understanding the MIDI standard Decoding MIDI files Psychoacoustics and Lossy Digital Compression Psychoacoustics and human hearing. MPEG Audio Layer Standards (MP3, AAC, etc)
51	Refererences Proakis, J. & Manolakis D., Digital Signal Processing, Prentice Hall, 1996; ISBN 0-13-373762-4 Watkinson, J., The Art of Digital Audio, Focal Press, 2005; ISBN 0-240-51587-0 Oppenheim, A. & Schafer, R., Discrete-Time Signal Processing, Prentice Hall, 1989; ISBM 0-13-216292-X http://www.harmony-central.com/Effects/effects-explained.html O’Connor, K., The Ultimate Tone Vol. 1 http://www.geofex.com/fxtech.htm http://www.freeinfosociety.com/electronics/schematics/audio/ringmodulator.html