Linear Time Invariant (LTI) Systems

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Helena Barber
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1 Linear Time Invariant (LTI) Systems Superposition Sound waves add in the air without interacting. Multiple paths in a room from source sum at your ear, only changing change phase and magnitude of particular frequencies. NEW FREQUENCIES DO NOT APPEAR Mixing audio channels electronically is similarly done by simple addition. Phasor Representation of Sinusoids (Complex Exponentials)

2 Time and Frequency Domains Convolution in the time domain is multiplication in the frequency domain. Practically, audio filtering is generally done in the time domain, but is better displayed and understood in the frequency domain. We sometimes display both positive and negative frequencies (phasors spinning forward and backwards).

3 Continuous vs. discrete time domain. Analog filters (capacitors, resistors, and coils). Pre-amp generally cut-off low frequencies and always cut-off high frequencies (to avoid sampling artifact). Power amps are also analog of necessity. Everything after A/D and before D/A is done in the digital domain.

4 Aliasing CD quality sound is sampled at 44.1 KHz, more than twice the range of human hearing.

5 Filters Linear Time-Invariant (LTI) systems are governed by Linear Differential Equations In the continuous domain. In the discrete domain this can be viewed as: 1 sample defines x [t] 2 samples defines x [t] 3 samples defines x [t] Finite Impulse Response (FIR) digital filters output is delayed and scaled samples from input. y[ n] = 2x[ n]+ x[ n 1]+ x[ n 2]+ 2x[ n 3] Ambience, reverb, with limited die-out time, Fourier, stable.

6 y[ n] = x[ n] x[ n 1] High-pass, differentiation favors high frequencies (which have steeper slopes) y n = x n + x n 1 + x n 2 + x n 3 Low-pass, moving average, box-car filter, integration favors low frequencies (which have larger areas)

7 Infinite Impulse Response (IIR) digital filters Output is delayed and scaled samples from input and output y[ n] = 4x[ n]+ 1 2 y [ n 1 ] Exponential decay, active systems (with energy) Laplace, can be unstable, sharper than FIR using less computation.

8 Frequency Shifting Multiplication in the time domain is frequency shifting in the frequency domain. Multiplication by a phasor.

9 Effects Tremolo (varying the volume) Easy to do with photo-resistor ( Light Dependent Resistor LDR in figure) and neon (fast) light bulb powered by Low Frequency Oscillator (LFO). Photo-resistor is linear, does not distort s beach guitar sound SV1 CLAV Variation 1, use the editor to show it, and the Spread knob to pan it left and right. Vibrato (varying the frequency) Rotating (Leslie) speaker Doppler shift. Now done digitally by frequency shifting. Chorus effect The SV1 CLAV Variation 1, the editor lists it as Organ Vib/Chorus The chorus effect in acoustic music is achieved by having more than once voice singing (nearly) the same note in a chorus, more than one string in a piano, 12-string guitar, or mandolin. Electronically, it can be produced by shifting the frequency slightly between multiple tone sources. Phasing (Flanging) Flanging is produced by mixing two identical signals together, one signal delayed by a small and gradually changing period, usually smaller than 20 milliseconds. This produces a swept comb filter effect: peaks and notches are produced in the resultant frequency spectrum, related to each other in a linear harmonic series. Varying the time delay causes these to sweep up and down the frequency spectrum. The "flange" effect originated with a tape recording technique using two tape machines, when an engineer put a finger on the flange, or rim, of one of the tape reels so the machine was slowed, slipping out of sync by tiny degrees with the other

(from Wikipedia) Other phase delay devices are also available. Originally used a speaker and microphone in isolated room.

10 Reverb machine. John Lennon came up with the name flanging during the recording of Abbey Road. More generally flanging is a type of phasing, namely, when the phase shift is proportional to frequency, yielding a constant time delay across all frequencies. (from Wikipedia) Other phase delay devices are also available. Originally used a speaker and microphone in isolated room. Analog circuit including actuator and pickup with springs hung between them. Do not bump! Later, progressed to tapped analog and then tapped digital delay lines. Echo Originally used tape loops with separated playback and record heads. Now easily done digitally.

11 Compression Intentional linear adjustment of volume. Automatic Gain Control (AGC), to keep perceived volume the same. Dolby Noise Reduction: reduced noise in analog recording and transmission systems by boosting quiet parts. Limiter Similar to a compressor, except that the limiter compresses only signals that exceed the specified level to lower unnecessary peak signals. Distortion: Intentional non-linear treatment of signal. Clipping, hard (digital, solid state) vs. soft (vacuum tubes). Introduces new harmonics to single pitch signals. Introduces new pitches to multi-pitch signals, because pitches interact nonlinearly Overdriving speakers, combines nicely with intentional runaway feedback (Hendrix). Causes sustain, as note takes longer to die out before get dropping below the clipping point. Fuzz box in 1960 s using non-linearity of diodes.

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