• Definition
• Uses

• Frequency Domain
• Time Domain

• Frequency Conclusions
• Time Conclusions

• Work Distribution
• C.J. Ganier
• Randall Holman
• Julie Rosser
• Erik Swanson

          For a CD, the sample rate is 44.1 kHz which makes the highest 
     frequency without aliasing 22.05 kHz.  Also, we noted that the
     average peak frequency that an adult can hear is approximately 18
     kHz, which leaves 4 kHz of frequency that is practically
     "unused".  We put this frequency range into use.

	  We begin with two signals, such as the ones below.  These 
     will be called the "base", which contains the hidden message, and 
     the "message", which will be hidden in the base.  The
     spectrograms of two such signals are shown below.

          Next, we take the base signal and lowpass filter it to 18 kHz,
     which will clear up the upper 4 kHz.  Following the example of
     the telephone company, we will bandpass filter the message signal 
     from about 300 Hz to 3.3 kHz, which will be a small enough band
     to fit in the upper portion of the filtered base.  The two
     filtered signals are shown below.

          Now, recalling techniques from ELEC 241 and what we've
     learned this semester, we modulate the filtered message using a 
     cosine with carrier frequency 20 kHz, the midpoint of our 4 kHz
     band.  We then combine the modulated, filtered message with the 
     filtered base signal and we get a signal with a hidden message in 
     it.  The spectrogram of the combined signal is shown below.

          So, what do we do now? If we are doing digital watermarking, 
     we are pretty much done.  A record label could encode a message
     in their songs, making it write-protected and un-recordable. But, 
     what if we worked for a government agency which wanted to relay a 
     secret message using our above method? How would you recover the
     hidden message?

          First, we must get rid of portions of the base signal that
     are in the combined signal.  We do this by bandpass filtering the 
     combined signal to above and below the hidden message.  Then, we
     modulate the message back down, using a cosine with the same
     carrier frequency as before.  Also, due to additive noise when we 
     did the addition and higher frequencies due to modulation, we
     must lowpass filter the demodulated message. We then have a
     signal which very closely resembles the sent message.  The
     results of this process can be seen in the recovered signal below.

          This process is a fairly simple one, requiring only
     filtering and amplitude modulation in order to hide the signal in
     the base.  It is very much a physically realizable process and
     one that we could accomplish with our current knowledge of
     electrical enigeering techniques.  This ease of implementation
     will more than make up for the small amounts of distortion in the 
     combined signals, as well as the limited frequency range of the 
     recovered signal.