DAC Output:
The level drops at the higher frequency?
Although sophisticated DACs such as over-sampling DACs or one-bit DACs are the trend now, let me talk about traditional DACs which directly output analog voltages according to the digital input. Suppose that a sine wave with a constant amplitude and frequency is applied to a traditional DAC. Since a sine wave in digital is a discrete signal, the signal level is defined only at each sampled point, or no levels are defined at any other point.
The signal levels are defined at points but they are not actually effective since points do not have any width. Here, turn on the "DAC Output." What you will see is the output of traditional DACs. It keeps the levels until the next sampled point. When the rise time is short comparing with the sampling period, the shape is like aligned rectangles. Next, turn on the "Sin Wave." When the frequency of the sine wave is much lower than the sampling frequency, there would be no problem but the difference between the real sine wave and the aligned rectangles becomes significant when the frequency is higher.
The difference is the cause of the topic of this article, that is, DAC output level drops at the higher frequency. Theoretically speaking, the rectangles or representing a sine wave by aligned rectangles make the level drop. As mentioned above, real digital sine waves consist of pulses without the width, not a set of rectangles. The sine wave consisting of aligned rectangles is convolution (Note 1) of the digital sine wave with a unit rectangle whose width is the sampling period. Since the convolution in the time domain is equivalent to multiplication in the frequency domain, the frequency characteristic of the sine wave consisting of aligned rectangles is the frequency characteristics of the original sine wave multiplied by the rectangles.
The frequency characteristic of the unit rectangle is represented by the following formula. Therefore, DAC outputs consisting of aligned rectangles are the multiplication of the original signal by the following formula.
The formula is called "sin(x)/x" because of the figure. The following is the representation of the formula where Ą= 1/Fs.
It is easily found that the gain at the Fs is zero. We need the response in the area lower than Fs/2. Here, if you put 0.4 on the "Gain at" and click the "Calc." you will see the gain is -2.420070[dB]. The maximum frequency of compact disc, 20KHz, is 0.453 of the sampling frequency 441.KHz. Input 0.453 in "Gain at" to calculate the gain, you will get -3.16dB as the gain. This cannot satisfy the typical tolerance -3dB. So you need some measures to keep the criteria.
(Note 1) Convolution is method often used in digital signal processing and the mathematical representation is the following.
When the signal is aligned rectangles, h(v) is one during a sampling period and zero for all others.
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