Sampled Digital Signals

Digital modulation is wholly dependent on the fundamental concepts of sampling theory. The subject of sampling is far too broad to be covered here, but a brief overview is given for the sake of clarity.

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Since the topic is modulation, we will use a sinusoidal signal as a model. A continuous time representation of a sinusoid is shown graphically in figure 2a. At any instant, “t,” on the horizontal axis, the amplitude of the sinusoid may be found on the vertical axis. A uniformly sampled version of the sinusoid is shown in figure 2b. Note that the amplitude is only known at certain discrete points in time (the sampling instants), which are uniformly distributed in time. Sampling theory states that as long as at least two sampling instants occur within a complete cycle of the sampled sinusoid, then the sinusoid can be completely reconstructed from the two sample points.

What makes sampling attractive is that the amplitude of the sample values can be encoded as twos-complement binary numbers. So, from a digital perspective, if we can generate a proper sequence of numeric values, we can generate a digital sinusoid. But why would we want to generate a digital sinusoid in the first place? Recall that modulation requires a carrier signal; i.e., a sinusoid. In the analog world this is implemented with an oscillator circuit that operates at the specified frequency. In the digital world, some form of a digital oscillator must be realized. It turns out that this can be readily accomplished with a numerically controlled oscillator.

Numerically Controlled Oscillator

In its simplest form, a NCO consists of a lookup table made up of sinusoidal sample values, (usually implemented as a read only memory — ROM), a binary counter for addressing the ROM, and a clock signal to drive the counter (see figure 3). Successive address locations in the ROM contain the successive sample values of the desired sinusoid. As the counter is clocked, each new count addresses the next ROM location causing the appropriate digital number to appear at the ROM output. The rate at which the counter is clocked is the sample rate of the system. If we examine the output of the ROM over time, we would observe a series of numbers that update at the sample rate. The numbers would span a numeric range that depends on the bit-width of the ROM output. Thus, the number of bits in the ROM's output word determines the resolution of the desired digital sinusoid. As an example, if the ROM output were 10 bits, then using twos-complement representation would yield a numeric range of -512 to +511 for the amplitude values of our sampled sinusoid.

The particular NCO example just described would only be capable of generating a digital sinusoid of one specific frequency; namely, the sample rate divided by the number of samples stored in the ROM (assuming that the samples stored in ROM span a single cycle of the sinusoid). A more flexible NCO would use a fairly large ROM (perhaps containing 4,096 samples, or more) and a counter that can count by some input modulus; that is, count by 1, 2, 3, 4, 5, etc. as determined by a “frequency control number”. For example, if the sample rate is 10 MHz, the ROM is 4,096 words in length, and the frequency control number is 1, then the output sinusoid would have a frequency of: 10 MHz/4096 or 2.44 kHz. If the frequency control number is 5, then the counter jumps 5 steps for each input clock period. This, in turn, causes every 5th ROM address to be accessed. The net result is a sinusoid with coarser amplitude steps, but of greater frequency. Specifically, the frequency of the new sinusoid would be: 10 MHz/(4096/5) or 12.21 kHz. In general, this can be expressed as: fs(N/M) where fs is the system sample rate, N is the frequency control number, and M is the length of the ROM.

Many variations on this theme can be found in the literature. The main point is that a NCO serves as a means for generating a digital sinusoid at a specific sample rate but with programmable frequency. The frequency is restricted, however, to an integer multiple of the sample rate divided by the ROM word length up to a maximum of ½ of the sample rate (the Nyquist constraint). The larger the ROM length (address range), the finer the frequency resolution. The more bits in the ROM output word, the finer the amplitude resolution.