interpolated delay lines ideal bandlimited interpolation...
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Interpolated Delay Lines
Ideal Bandlimited Interpolation
Fractional Delay Filter Design
Low-Order (Fast) Interpolators Linear
Allpass
High-Order Interpolation Ideal Bandlimited Interpolation
Windowed-Sinc Interpolation
Optimal FIR Filter Design for Interpolator
Least Square
Summary
Simple Interpolators for Real-Time
Fractional Delay Filtering
Linearly Interpolated Delay Line (1st-Order
FIR)
Allpass Interpolated Delay Line (1st-Order)
Frequency Responses of Linear
Interpolation for Delays between 0 and 1
Linear Interpolation as a Convolution
x [n ] x̂[n− Δ ]
Phase Delays of First-Order Allpass
Interpolators for Various Desired Delays
First-order allpass interpolation
Windowed sinc
Effects in
time-space domain
Summary
Fractional Delay line – overview
Time Variant Fractional Delay line
Digital effects
Industry standard structure
Vibrato
Flanger
Chorus
Eco
Leslie
Delay line – Overview
Minimum delay time is depending on sample rate:
Tmin = 1/Fs
We define Tmin as the unitary delay time
In many applications it’s needed a “continuous delay”
Fractional Delay Line
z− k y[n]= x[n− k ]x [n ] k ∈ Z
Fractional Delay line – Overview
We define D
D can be defined as the sum of an integer part and a real part
0<= α< 1
Defining an operator LD, we obtain
Where LD is the delay operator and an interpolation operator which can be a function or a filter
z− D y[n]= x[n− D]x [n ] D∈ R
Interpolation
Filter
Time Variant Fractional Delay Line
Many digital effects are based on a Delay variant on time
Implementation
Time Variant Fractional Delay Line
y[n ]= x[n− D[n ]] D depends on n (time)
Time Variant Fractional Delay Line
We write as
D0 is the nominal length of the delay
fd[n] is the variation function
is the variation index
the type of the effect is depending on fd[n] and md
A sinusoidal variation function is often used (LFO)
Considering the original relation, we obtain
D0 is the mean values of the delay line
D[n ]
y[n] = x[n−D0(1+mD fD [n])
mD=D1
D0
∈ [0,1]
Time Variant Fractional Delay Line Phase Modulation by TV-FDL
Given xm[n] a modulant signal and x[n] the carrier
We define modulation as
where f is a generic modulation function, if it is true that
where g is the demodulation function
The Phase Modulation can be written as
if D[n] = kpxm[n] Phase Modulation made
by TV-FDL
Depending on the modulation function we will also have a modulation in frequency.
Traditional effects
Traditional effects are typically based on the superposition of multiple delayed version of the musical signal Depending on delays between repetitions we have
Echo
Chorus
Flanger
Based on modulated delay line Which also allows us to make
Vibrato
Doubling
Industry standard structure
Dattorro proposed a general scheme for some digital effects
Modulated delay line:
delay D[n] is modulated (LFO)
feedback fb
Feedforward ff
Output: modulated signal mixed with the non-modulated one and
controlled with a blend coefficient b
Z− D[n]
LFO
Vibrato
Vibrato is a simple frequency modulation
Remove feedback and blending (only modulated
output signal)
Delay line sized for a delay of about 5 ms
A minimal delay (below 1ms) gives best results
Z− D[n]
b ff fb
Z− D[n]
Flanger
Add to the original signal a “dynamically” delayed signal Delay must be within the ear’s integration time
A 1 ms delay line is OK for most applications
This results in a richer spectrum (FM)
The above coefficients tend to maximize the “spectral comb”
Allpass interpolator is preferable The amplitude attenuation of a linear interpolator would damage
high-frequency peaks
b ff fb
Standard chorus
Add to the original sound a slightly delayed replica of its (a 5 ms delay is OK for most cases) In this case the spectral comb is undesirable, therefore we remove the feedback
Transparency is usually desirable (guitarists usually prefer) Linear interpolation is not suitable because of high-frequency attenuation
A better solution (white chorus) is to feed the signal back, while keeping the whole system as allpass as possible, i.e. setting fb=b. This way the output is richer at high frequencies without using an allpass
Residual peaks is reduced by setting ff different from b
b ff fb
Z− D[n]
Z− D[n]
b ff fb
Doubling
It’s a chorus with a single delay line and Depth
values generally higher
Mostly used by singers as it doubles the voice
Delay margin can change quite significantly (typically 20
ms)
Modulation should be rather random
Echo
Needs a significantly long delay line
below 80 ms we cannot distinctly perceive two replicas of
identical signals
coefficients must be adjusted depending on desired timbre
b ff fb
Z− D[n]
Electro-mechanical ancestor Rotary Speaker – Leslie
One of the most famous effect (Electro-mechanical) use to effect
the sound of the Hammond
Two rotary horn as treble
speakers (only 1 active)
One rotary cylinder for bass
speaker
Possibility of changing rotary
speed
Choral (low) ~ 15-20 rpm
Tremolo (high) ~ 300-500
rpm
Leslie In model 122 only one treble rotary speaker is used
Wood rotary cylinder in front of the woofer
If the listener is in a fixed position
Amplitude modulation given by the high directivity of the rotating speaker
Pitch modulation given by the doppler effect
The max amplitude value will be when
The max pitch modulation will be when
Leslie Generally Leslie it’s amplified with 3 microphones:
a stereo couple for the treble
high directivity
a single mic for the bass
doppler effect is less evident
Leslie Considering only the rotating horns
Leslie
2 channels: yL[n] and y
R[n]
Frequency modulation for doppler effect (TV-TDL)
Amplitude modulation for each channel