What is the maximum count interval using a 12 -stage counter operated at a clock rate of \(20 \mathrm{MHz}\) ?

A digital counter is an electronic device or circuit that counts the number of occurrences of an input signal. It typically consists of a series of flip-flops, which are binary devices that can exist in one of two states, usually denoted as '0' or '1'. These flip-flops are connected in a way that allows them to represent a binary number. With each pulse of the input signal, known as the clock signal, the counter increments its binary value by one.

For a 12-stage digital counter, as mentioned in the original exercise, there are 12 of these flip-flops connected in series. This setup enables the counter to represent a binary number up to 12 bits in length. The number of unique values a counter can represent is called its 'count capacity' or 'modulus.' A 12-bit counter has a count capacity of \(2^{12}\) states, which translates to 4096 distinct states each representing a unique binary number from 0000 0000 0000 to 1111 1111 1111 in binary representation.

Clock frequency, often measured in hertz (Hz), is a key concept in digital electronics. It denotes the number of pulse periods per second that a clock signal generates. For digital counters, this clock signal acts as a time reference that triggers counting operations. The faster the clock frequency, the shorter the time between count increments.

In the context of the given problem, the clock frequency is specified as 20 megahertz (MHz), which means that the counting device receives 20 million pulses every second (\(20 \times 10^6\) pulses per second). This high rate of pulses allows the counter to iterate through its states rapidly. Understanding the relationship between the clock frequency and the counter's operation is crucial because it determines how quickly the counter goes through all of its binary states and, hence, how fast it can count.

The concept of binary states underpins the functioning of digital counters. A binary state refers to one of the two possible states, often represented by '0' (off) or '1' (on), that each stage of a counter can achieve. The binary system is the foundation of digital electronics, as it is a simple yet powerful means of representing and processing data.

In binary, each digit's position represents a power of 2, starting from \(2^0\) on the rightmost side. For the 12-stage counter in our exercise, each stage can be in one binary state at any one time. When all stages are put together, they can represent a large combination of binary states — precisely \(2^{12}\) or 4096 different combinations. The ability to understand and calculate binary states is essential for students dealing with digital electronics because it directly affects the design and functionality of digital circuits such as counters.