• in previous posts, calculating the spectrum of signals was explored
    • in most cases, this spectrum is not wideband: it is mostly limited around a certain range of frequencies
  • depending on this range, different types of signals can be defined.
    • if most of the energy of a signal is concentrated around zero (resp. \(-\pi\) or \(\pi\)), it is a lowpass (reps. highpass) signal
    • if the energy is concentrated somewhere in between, it is a bandpass signal

signal modulation

  • fourier transform modulation theorem:
    • allows to transform a signal
  • for example, a lowpass signal can be modulated into a bandpass signal
    • this operation can be reversed by demodulating
  • this is obtained by simply multiplying the signal by a cosine at the adequate frequency
  • having seen this theoretical result, it can be put to work on a practical application like tuning a guitar

categories based on energy concentration

  • there are three broad categories according to where most of the spectral energy resides
    • lowpass signals (baseband signals)
    • highpass signals
    • bandpass signals

  • lowpass signal: energy is mostly concentrated around the origin

    lowpass

    fig: lowpass signal

  • bandpass signal: energy is mostly concentrated around \(-\pi\) and \(\pi\))

    lowpass

    fig: bandpass signal

  • highpass signal: energy is mostly concentrated around \(\frac{-\pi}{2}\) or \(\frac{\pi}{2}\)

    lowpass

    fig: highpass signal

sinusoidal modulation

  • this type of modulation is obtained by multiplying a signal \(x[n]\) with a \(\cos(\omega_c n)\)
    • \(\omega_c\) is the carrier frequency

  • to analyze the spectrum of this modulation, take DTFT: \[ \begin{equation} DTFT\{ x[n]\cos(\omega_c n) \} \\ = DTFT \bigg \{ \frac{1}{2} e^{j \omega_c n} x[n] + \frac{1}{2} e^{-j \omega_c n} x[n] \bigg \} \
    = \frac{1}{2} \bigg [ X(e^{j(\omega - \omega_c)} ) + X(e^{j(\omega + \omega_c)}) \bigg ] \end{equation} \]

  • modulation, pictorially:

    sin-mod-0

    fig: begin with source signal

    sin-mod-1

    fig: apply \(\omega_c\) shift

    sin-mod-2

    fig: apply \(-\omega_c\) shift

    sin-mod-3

    fig: modulated signal

  • when modulation frequency is too large:
    • when \(-\omega_c\) is close to \(\pi\) or \(-\pi\)
    • the original signal loses shape and information is lost

application of modulation

  • modulation brings the baseband signal to the transmission band
    • i.e. voice to radio frequencies
  • demodulation at the receiver brings it back
    • i.e. radio to voice
  • voice and music are lowpass signals
    • energy is lost during transmission over very short distances
  • radio channels are bandpass signals
    • their modulation frequencies are higher, else they lose the information embedded in them through interference
  • radio waves are carrier signals and are modulated with audio sources
  • then, they are transmitted from source to destination
  • at the destination, the source audio is retrieved from the carrier by demodulation

sinusoidal demodulation

  • simply multiply the received signal by the carrier again to get the original signal \[ y[n] = x[n]cos(\omega_c n) \] \[ Y(e^{j \omega}) = \frac{1}{2} \big [ X(e^{j(\omega - \omega_c)}) + X(e^{j(\omega + \omega_c)}) \big ] \]

\[ \begin{equation} DTFT \{ y[n]\cdot2\cos(\omega_c n) \
= Y(e^{j (\omega - \omega_c)}) + Y(e^{j (\omega + \omega_c)}) \
= \frac{1}{2} \big [ X(e^{j(\omega - \omega_c)}) + X(e^{j\omega}) + X(e^{j\omega})+ X(e^{j(\omega + \omega_c)}) \big ] \
= X(e^{j\omega}) + \frac{1}{2} \big [ X(e^{j(\omega - \omega_c)}) + X(e^{j(\omega + \omega_c)}) \big ] \end{equation} \]

  • demodulation, pictorially:

    sin-demod-0

    fig: source signal

    sin-demod-1

    fig: modulated version of signal

    sin-demod-2

    fig: signal shifted to right

    sin-demod-3

    fig: signal shifted to left

    sin-demod-4

    fig: sum of shifted signals

    sin-demod-5

    fig: demodulated signal

  • the baseband signal can be recovered
  • but some spurious high-frequency components exist
    • those will have to be filtered out

application: guitar tuning

  • problem statement:
    • reference sinusoid: frequency \(\omega_0\)
    • tunable sinusoid: frequency \( \omega \)
  • tuning:
    • make \( \omega = \omega_0 \) “by ear”

procedure

  1. bring \(\omega\) close to \(\omega_0\)
  2. when \(\omega \approx \omega_0\), play both sinusoids together
  3. trigonometry can then be used:
    • \( x[n] = \cos(\omega_0 n) + \cos(\omega n) \)
    • \( = 2 \cos(\frac{\omega_0 + \omega}{2} n) + \cos(\frac{\omega_0 - \omega}{2} n) \)
    • \( \approx 2 \cos(\Delta_\omega n)\cos(\omega_0 n) \)

procedure analysis

  • in \( x[n] \approx 2 \cos(\Delta_\omega n)\cos(\omega_0 n) \)
    • error signal: \( 2 \cos(\Delta_\omega n) \)
    • modulation at \(\omega_0\) \( \cos(\omega_0 n) \)
  • when \(\omega \approx \omega_0\), error is too low to be heard
    • so the modulation signal multiplication brings it up to hearing range
    • it is perceived as amplitude oscillations of carrier frequency
  • pictorially:
    • red signal is the carrier frequency
    • blue is the audible beats heard

    tuning-0

    fig: time domain signals - beat frequency

    tuning-1

    fig: time domain signals - slower beat frequency

    tuning-2

    fig: almost nil beat frequency