Time shifted unit impulse function in frequency domain

Question

How can we represent a time shifted unit impulse function in frequency domain ? Fourier transform of unit impulse function is 1. So I think it will be an infinite parallel line in frequency domain. What about for the time shifted version ?

Answer 1

Apply the Time Shifting property of Fourier Transform.

If $$ x(t) \implies X(j\omega) $$ then $$ x(t-a) \implies e^{-j\omega a} X(j\omega)$$ For impulse, $$ \delta (t) \implies1$$ Means, it contains all frequencies and their magnitudes are unity.

For the time shifted version, $$\delta(t-a) \implies e^{-j\omega a}$$

This is a complex number with magnitude = 1. Hence the magnitudes of all the frequency components are still unity in the frequency domain. But their phase angles are different, $$ \phi = -\omega a $$ In the earlier case, the phase \$ \phi \$ was zero for all the frequency components.

Answer 2

The magnitude of the FFT is 1 for all frequencies, but don't forget that the result is a complex function — there's a phase angle component, too. For the non-shifted pulse, those angles are all zero, but when you time-shift the pulse, the phase angles for different frequencies become different.

Answer 3

We expect the same magniitude but phase of spectrum for an impulse will integrate down from 0 to -2π at 1/f=T for a given time delay T. It will then be recursive from 0 to -2π repeating at intervals of f=1/T in the spectrum.

We expect the same phase negative integral ramp for a delayed Step Response.

I found (that) understanding this feature in class to be useful 10 yrs later, when I was investigating an EMI problem. (explained below picture)

(real world example)

What if the impulse is just a pulse of some amplitude and some pulse width?

• It is a step with an delayed inverted step then both repeated
• Can you see how the spectrum of the pulse drops below -60dB or null here?
• That is your time delay or width of the pulse. You will not find any sine wave energy at that frequency = f=1/T or any harmonic of that 2/T,3/T.
  • Since my rise time was finite the recursive humps decay with rising f.
  • In the simulation above I used 10 Hz rep. rate and hand drew a pulse on this webpage and displayed the log amplitude and phase of the FFT.
• in my real world problem, I discovered someone installed an air deionizer in the ceiling producing 30kV DC on the ceiling outside the cleanroom with electrode pins insulated a 2 cm metal cup with alternating impulses to deionize the airborne dust.
  • unfortunately it produced continuous spectrum to up the antenna wavelength of the arc pin gap and then repeated which then interfered with my Servo Writer.
    We had mysterious errors in a delicate magnetic recording machine called a SERVO WRITER that writes impulse currents on a rotating disk later used inside a hard disk drive for servo positioning. When I examined the fault, I recognized stray RF noise impulses in the narrow band frequency (~10MHz) but occurring at a low repetition rate like < 10 pulses per second. I discovered the source of interference outside the Class 100 Clean room from an air deionizer installed in the hallway. It was installed outside the cleanroom at a Burroughs disk drive factory in the mid-80's to reduce the airborne dust contamination outside the room in the hopes that translated to less dust being transported past the air shower and inside the clean room.

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