How To Convert This Spectrum To Time Domain
Question
Given a spectrum with:
- amplitude
+1atf = -100 Hz - amplitude
-3atf = 0 Hz - amplitude
+1atf = +100 Hz
what is the corresponding signal in the time domain?
Short Answer
[ x(t) = -3 + 2\cos(2\pi 100 t) ]
Why
1. The f = 0 component
The line at 0 Hz is a DC component, so it gives a constant term:
[ -3 ]
2. The symmetric lines at \pm 100 Hz
Equal spectral lines at +f_0 and -f_0 correspond to a real cosine term.
In general,
[ A\cos(2\pi f_0 t) \quad \leftrightarrow \quad \frac{A}{2}\delta(f-f_0)+\frac{A}{2}\delta(f+f_0) ]
Here each side has amplitude 1, so:
[ \frac{A}{2}=1 \Rightarrow A=2 ]
and the oscillating part is:
[ 2\cos(2\pi 100 t) ]
3. Combine both parts
Therefore:
[ x(t) = -3 + 2\cos(2\pi 100 t) ]
Time-Domain Interpretation
This is a cosine wave with:
- mean value
-3 - amplitude
2 - frequency
100 Hz
So it oscillates between:
[ -3 + 2 = -1 ]
and
[ -3 - 2 = -5 ]
Counterpoints and Gaps
- this interpretation assumes the shown spectral values are real-valued line amplitudes with zero extra phase information
- if additional phase information were present, the time-domain expression would need an added phase shift