1. Lecture 3
From Delta Functions
to Convolution

2. Recap…. From Fourier series to Fourier transforms Pulses and top hats
“Wide in time, narrow in frequency; narrow in time, wide in frequency”
Bandwidth

3. Outline of Lecture 3
Delta functions, conjugate variables, and uncertainty
Magnitude and phase
Power spectra
Convolution

4. Time-limited functions and bandwidth
HWHM ~ 25 Hz

5. Fourier Transforms and the Uncertainty Principle
Note the ‘reciprocal’ nature of the characteristics of the function and those of its Fourier transform.
Narrow in time  wide in frequency: Dt   Df 
Compare this with
DxDp ~ (h/2p)/2
From the Fourier transform of a Gaussian function we can derive a form of the uncertainty principle. (Further Exercises, Problems Class 2)

6. The ultimate time-limited function: Dirac d-function?
So, in the limit of the pulse width → 0, what happens to the pulse’s Fourier transform?
Fourier transform becomes broader and broader as pulse width narrows.
In the limit of an infinitesimally narrow pulse, the Fourier transform is a straight line: an infinitely wide band of frequencies.
Dirac delta-function, d(t): ?
Calculate the Fourier transform of d(t).

7. Magnitude, phase, and power spectra
Fourier transform is generally a complex quantity.
- Plot real, imaginary parts
- Plot magnitude
- Plot phase
- Plot power spectrum:|F(w)|2
Take z = e-ikx ?
Write down the magnitude (or modulus) of z.

8. Some revision…
You should be able to answer the following questions (Q2 of last week’s problems class question sheet). ?
Calculate the Fourier transform of d(t-t0). ?
How is the magnitude of the Fourier transform affected by the shift in the function? ?
How are the phases affected?

9. Why ‘power’ spectrum?
The power content of a periodic function f(t) (period T) is:
If f(t) is a voltage or current waveform, then the equation above represents the average power delivered to a 1 W resistor.
Parseval’s theorem states:

10. Why ‘power’ spectrum?
For aperiodic signals, Parseval’s theorem is written in terms of total energy of waveform:
Total power or energy in waveform depends on square of magnitudes of Fourier coefficients or on square of magnitude of F(w). (Phases not important).

11. The importance of phase…
Karle
Hauptmann
Karle and Hauptmann won the Nobel prize in 1985 for their work in X-ray crystallography.
Take Fourier transforms of images to left. A B
Mix phase spectrum of Hauptmann with magnitude spectrum of Karle (A) and vice versa (B).

12. Measurement & Convolution
Convolution underlies every measurement we make.
It’s what, for example,
turns this:
…into this:
Hubble space telescope – spherical aberration led to image blur

13. Measurement & Convolution
‘Convolution’, n. Origin: Latin: from ‘convolvere’ – ‘roll together’
Every measuring instrument/device is associated with a finite
resolution – model this with an impulse response function.
Very simple concept: apply a delta function input to the system – what is the output (i.e. the system response)?

14. Measurement & Convolution
The system could be:
- an electronic circuit
- a camera or imaging system [point spread function]
- a mechanical system (eg. suspension on a car)
- an audio amplifier
- a spectrometer
Vast range of applications (from spectroscopy to audio systems to forensic science…..)
Wen et al., For. Sci. J (2002)

15. Convolution in Optics: The Point Spread Function
Delta-function at ‘input’ forms finite sized spot at ‘output’ – the impulse response function is the point spread function (PSF)

16. Convolution in Optics: The Point Spread Function
Original image
‘Output’ image

17. So how do we convolve two functions?
The convolution of two functions f(x) and g(x) (f  g) is given by:
OK, but what does that integral actually do…?
1. Sketch f(x)
2. Sketch the other function (g(x)) backwards on a transparency
3. Incrementally slide the transparency across the graph paper in the +x direction
4. At each point (x’) calculate the area under the curve representing the product of the two functions.

18. Impulse Response and Convolution
Impulse Response and Convolution ?
This is the impulse response (r(t)) for a mechanical system.
(i) Write down a mathematical expression to describe this response function;
(ii) Sketch the system response for an impulse applied at time t0 (> 0)

19. Convolution and spectra
Convolution and spectra

20. So how do we convolve two functions?
Is there not an easier way of convolving two functions? The integral seems tricky to calculate and the graphical method is laborious.
Convolution theorem:
The Fourier transform of the convolution of
two functions is 2 times the product of
the Fourier transforms of the individual functions:
FT (f  g) = 2 F(k)G(k)
Extremely powerful theorem

21. Impulse response and convolution.
The response of a system (optical, audio, electrical, mechanical, etc..) to an arbitrary signal f(t) is the convolution of f(t) with the impulse response of the system.
f(t) may be represented as a series of impulses of varying height. System responds to each of these in a characteristic fashion (impulse response).
f(t)
To get response to ‘stream’ of impulses (i.e. f(t)) convolve f(t) with impulse response function. t

22. Impulse response and convolution.
Can also deconvolve if we know the impulse response (or point spread) function. (HST before corrective optics). ?
How do you think it was possible to evaluate the point spread function for the Hubble telescope?

23. Impulse response and convolution: Audio signals
Remember that convolution holds for a vast range of systems.
Another example – audio signals.
Large concert hall
Ice cavern
Recording studio
Record impulse response of each environment. Then convolve with given signal to recreate charateristic acoustics of concert hall, cavern, or recording studio…

24. Impulse response and convolution: Audio signals
Now, take a recording….
and convolve this with the impulse response functions on the previous slide…