# Spectrum Inversion Problem

For a signal with $$f_c$$ (Centre Frequency) $$=1200MHz \,\,$$ and $$BW= 100MHz$$ ,which of the following Sampling Frequency ($$f_s$$) will cause spectrum inversion:
A)$$\, 287.5MHz \quad$$ B)$$\,575MHz \quad$$ C)$$\, 1150MHz \quad$$ D)$$\, 1600MHz \quad$$

My Thoughts:
As Centre Frequency ($$f_c$$) is mentioned in the question
$$\therefore$$ The given is Bandpass Signal (or Modulated Signal)

As Bandwidth ($$BW$$) of the given signal is $$100MHz$$
$$\implies (\text{Modulated signal})_{BW}=100MHz$$
$$\implies (\text{Baseband signal / Message signal})_{BW}=50MHz$$

In general,considering Triangular spectrum for Baseband Signal ($$X(f)$$)
then, Spectrum of Baseband Signal:

Now,In the question spectrum inversion is given
and we know spectrum inversion occurs only in FM (Frequency Modulation)
i.e odd multiples of lower sidebands of FM are inverted
For eg.:
If Baseband signal ($$x(t)$$)= $$\cos(2 \pi f_mt)$$
then, its spectrum is as follows:

Now, if this message signal undergoes frequency modulation,then modulated signal spectrum will be given as:

Here,we can see $$(f_c-f_m)$$ component is inverted
similarly,$$(f_c-3f_m)$$ component is inverted
and so on...

Now how can i proceed to find $$f_s$$,please help...

• Hi! What exactly does spectrum inversion mean ? odd multiples of lower sidebands of FM inverted still does not make sense (to me). – Fat32 Feb 13 '19 at 9:18

I don't think that FM has anything to do with this, unless you are using the term "spectrum inversion" in an unusual context.

The original signal is presumably a real signal centered at 1200 MHz with a bandwidth of 100 MHz. If you sketch its spectrum conceptually, it will have some content from 1150 to 1250 MHz, and a conjugate-symmetric version of the same thing from -1150 to -1250 MHz. The conjugate symmetry follows from properties of the Fourier Transform and the fact that the original signal is real. Call the portion of the spectrum at positive frequencies the positive image, and the portion of the spectrum at negative frequencies the negative image. If your sampling ends up reversing the positions of the positive and negative images, or if you convert the negative image to complex baseband, then that's spectral inversion.

To find out whether your a sampling scheme causes spectral inversion, recall that the spectrum of the sampled signal repeats at intervals of the sampling rate. Sketch this operation for the proposed sample rates and look at the position of the positive and negative images in the region from $$-f_2/2$$ to $$+f_s/2$$. Does the spectrum have the original sense, or has it been inverted?