Outline: Linear Systems Theory I

Fourier (spatial frequency) analysis of auditory or other one-dimensional signals

Signal: intensity as a function of time, f(t)

Signal representation:

Fixed basis set of signals

All others can be represented as a weighted sum of basis signals

Define: weight (scalar multiplication) = scale amplitude

Define: sum of signals= point by point sum of intensities at each time t

Pointwise signalrepresentation

Basis set is impulses: signals that are zero for all times except one instant, where the value=1

Weights to represent signal f are the amplitude values f(t)

Frequency representation

Basis functions are sine waves, defined by:

Frequency (cycles/second = Hertz = Hz)

phase (when the sine wave starts, time delay)

amplitude = intensity

Any signal can be so represented

Weights describe frequency content of the signal
The collection of weights and phases is called the Fourier transform

Examples

Square wave = sin t + (sin 3t)/3 + (sin 5t)/5 + (sin 7t)/7 + ...

Musical sounds are periodic

They have a repeating waveform with period T
They have a perceived pitch equivalent to a pure tone of frequency f = 1/T
The nonzero weights are only for frequencies

f (the fundamental), and
2f, 3f, 4f, ... (the harmonics)

The pattern of weights for the fundamental and harmonics determines the timbre

Linear systems analysis of systems

What is a system?

A box which takes one signal as input and outputs another
Examples:

Outer ear
Outer+middle ear
Outer ear through to a particular spot on the basilar membrane
Your amplifier and speaker system (well, at least the left speaker channel...)

Linear, shift-invariant systems

Linearity includes:

Homogeneity or the scalar rule

If input f yields output g, then input 2f yields output 2g (and likewise for 3f, 1.5f, etc.)

Additivity

If f₁ yields g₁ and f₂ yields g₂, the f₁ + f₂ yields g₁ + g₂

Shift-invariance

If input f yields output g, then input f delayed by 10 seconds yields g delayed by 10 seconds

If a system is linear and shift-invariant

We can summarize what it does to any input image by what it does to a small number of carefully chosen images

Example: pointwise image representation

Apply the system to a single impulse

The resulting image is called the impulse response

Any input is a sum of weighted, shifted impulses, and hence the resulting output is a sum of weighted, shifted impulse responses

Example: sine wave (Fourier) representation

Useful, nonintuitive fact: the response of a linear, shift-invatiant system to a

sine wave is the same sine wave again, with perhaps a

time delay (phase shift) and a scaling of amplitude. The phase

shift and amplitude scaling amounts will depend on the

frequency of the input sine wave

Therefore, for a given sine wave you know what the system does to it as two numbers:

amplitude scaling (called modulation transfer), and

phase shift.

Modulation Transfer Function (MTF):

Amplitude scaling as a function of frequency

If you know the MTF (and the phase shifts), then you know what the system does

to any image, since any image can be represented as a sum of sine waves

Example MTFs:

The bass control on your stereo

Controls the overall intensity at the output of a filter that emphasizes low frequencies

A low-pass filter

Your outer and middle ear

Emphasizes a range of frequencies important for speech perception (1-4Khz)

A band-pass filter