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• Modulation Review• Delay Issue

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In this example, consider using sine and cosine demodulation of cosine modulated signal x[n] where there is a delay. In, particular, for the case where

Consider a delay of 5 samples (so that 25*Omega_1*D = pi/4) in the modulated signal.

The following plots show the impact of a 5 sample delay on modulation and demodulation at four different modulation frequencies (but the same delay). The different modulation frequencies will change the phase shift introduced by the delay, and generate different outputs from sine and cosine demodulation. In particular, the modulation frequencies considered are 25, 50, 75 and 100 times Omega_1.


Plot of x[n] and X[k]

Note: 0.3 for coefficient 0 and ½*0.3 for coefficients -5, -4,-3 and 3, 4, 5


Cos25 Omega_1 n modulated x[n]

Note: ½*0.3 for coefficient 25 and -25, 1/4*0.3 for coefficients -30, -29,-28, 22,-21, -20 and 30, 29, 28, 22, 21, 20


Plots of modulated x[n] and x[n-5] (note periodicity!)


Cos25 Omega_1 n modulated x[n] after 5 sample delay

Note impact of pi/4 phase shift (peaks are 0.3*½ *sqrt(2)/2 for real and imaginary part).


Cos25 Omega_1 n demod after delay


Sin 25 Omega_1 n demod after delay


Cos50 Omega_1 n modulated s[n]

Note: ½*0.3 for coefficient 50 and -50, 1/4*0.3 for coefficients -55, -54, -53, -47,-46, -45 and 55, 54, 53,47,46, 45


Cos50 Omega_1 n modulation after 5 sample delay

Note impact of pi/2 phase shift (peaks are 0.3*½ but are now in the imaginary part).


Cos75 Omega_1 n modulated s[n]

Note: ½*0.3 for coefficient 75 and -75 and 1/4*0.3 for coefficients -80, -79,-78, -72,-71,-70, 80, 79, 78, 72,71, 70


Cos 75 Omega_1 n modulated s[n] after 5 sample delay

Note impact of approximately 3pi/4 phase shift (peaks are 0.3*½*sqrt(2)/2 for real and imaginary part, and opposite in sign from 25 Omega_1 case.


Cos100 Omega_1 n modulation

Note: ½*0.3 for coefficient 100 and -100, and 1/4*0.3 for coefficients -105, -104, -103, -97, -96, -95, 105, 104, 103, 97, 96, 95


Cos 100 Omega_1 n modulation after 5 sample delay

Note impact of pi phase shift (peaks are 0.3* ½ and real, but flipped in sign).

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