Unfortunately, the act of producing an analog signal from a digital one results in frequencies appearing that are much higher than the ones recorded, in addition to the appearance of frequencies across the spectrum. Therefore, the output needs to be filtered after converting it to analog for the speakers. With a good filter, we could filter the output signal again at 20 kHz to remove the high frequency noise, but low frequency noise will remain. But, filters are not perfect, and good filters are expensive with steep slopes in the cut-off region. Any part of the signal that has frequencies lying within the range of the steep slope of a filter will have a severe phase shift for those frequencies. An ideal filter will perfectly pass any part of the signal that is desired, and perfectly block any part of the signal that is not desired. In the frequency domain, this means continuously blocking certain frequencies and continuously passing others. What is desired is to block frequencies higher than can be heard and pass lower frequencies that can not be heard. This is where digitizing the signal introduces the first problem. When a signal is sampled and recorded, all the values for the amplitude of the signal are stored in quantized levels. Even though this signal has had all the frequencies above the Nyquist frequency removed initially, the analog reproduction of the digital signal will create high frequencies in the output, which will be aliases back into the signal without a good output filter. The input signal is represented by amplitude values which are separated from each other by a unit of time, ts . The signal is now represented by a series of values which are set at a discrete possibility (2^16) of values. Each sample point is given its own integral value, starting with zero. f[n] represents the entire signal. Since a sinusoidal term can be made into its equivalent exponential term ejw, each frequency component of the signal F(ejw ) is obtained from the sampled signal by the formula 8.
This formula repeats the frequency response every w = 2*pi for each value of n. This means that the frequency response for the signal ?reappears? at multiples of 2*pi. We need to remove these frequencies from the output, so they are traditionally filtered with another low-pass filter.
Quantization introduces an irreducible error, however. The measured value of the signal at any point is only an approximation. This error will propagate through the transformation of the digital signal to the output. With skill and luck, we should not introduce any other significant error to the signal. With a decent filter, we will smooth the signal output so the error is undetectable by the listener.