Question

Determine the total time it takes to transmit an uncompressed grayscale image (with 8 bits/pixel) from a screen with a resolution of \(1,280 \times 840\) pixels using each of the following media: a. A \(56 \mathrm{Kbps}\) modem b. A \(1.5\) Mbps DSL line c. A \(100 \mathrm{Mbps}\) Ethernet link

Short answer

Transmission times are approximately 153.6 s for the 56 Kbps modem, 5.7 s for the 1.5 Mbps DSL, and 0.086 s for the 100 Mbps Ethernet.

Step-by-step solution

Calculate the Total Number of Pixels

First, determine the total number of pixels in the image. The image has a resolution of \(1,280 \times 840\). Multiply these dimensions to find the total: \[ 1,280 \times 840 = 1,075,200 \text{ pixels} \].

Calculate Total Number of Bits

Since the image is grayscale with 8 bits per pixel, multiply the total number of pixels by the bits per pixel to find the total number of bits in the image: \[ 1,075,200 \times 8 = 8,601,600 \text{ bits} \].

Calculate Transmission Time for a 56 Kbps Modem

The modem speed is \(56 \text{ Kbps} = 56,000 \text{ bits per second}\). The transmission time \(t\) in seconds is given by the formula: \[ t = \frac{\text{Total bits to transmit}}{\text{Transmission speed}} \] Substitute the values: \[ t = \frac{8,601,600}{56,000} \approx 153.6 \text{ seconds} \].

Calculate Transmission Time for a 1.5 Mbps DSL Line

The DSL line speed is \(1.5 \text{ Mbps} = 1,500,000 \text{ bits per second}\). Using the same formula for transmission time:\[ t = \frac{8,601,600}{1,500,000} \approx 5.7 \text{ seconds} \].

Calculate Transmission Time for a 100 Mbps Ethernet Link

The Ethernet link speed is \(100 \text{ Mbps} = 100,000,000 \text{ bits per second}\). Again, applying the formula for transmission time:\[ t = \frac{8,601,600}{100,000,000} \approx 0.086 \text{ seconds} \].

Key concepts

Grayscale Image

A grayscale image is a type of digital image where each pixel represents a different shade of gray. Unlike color images, which use multiple channels—such as red, green, and blue—to convey colors, a grayscale image uses only one. This means each pixel can only store information about brightness. This makes grayscale images simpler and less data-intensive than their color counterparts.
Grayscale images are often used in applications where color information is unnecessary. For example, things like security cameras or certain types of medical imaging prefer grayscale due to the simplicity and focus on detail in contrast.

• A grayscale image uses only shades of gray.
• Each pixel has a singular value determining brightness.
• Less storage required compared to color images.
• Common in applications like security and medical imaging.

Examples of grayscale images are black-and-white photographs, where the absence of color allows for emphasizing texture and shapes.

Transmission Speed

Transmission speed refers to how fast data can be sent over a network. It's measured in bits per second (bps). Higher speeds mean data is transferred faster, reducing the time users wait for downloads or data exchanges.
The transmission media—such as modem, DSL, or Ethernet—determine speed capacity:

• Modem (like 56 Kbps): Slow and good for small data transfers. Common in early internet browsing.
• DSL (like 1.5 Mbps): Faster than modem, more suitable for modern internet tasks.
• Ethernet (like 100 Mbps): Very fast and ideal for large data transfers or real-time applications.

High-speed transmission can handle large files, like the 8,601,600 bits in a grayscale image, more efficiently, affecting the user's experience positively.

Pixels

Pixels are the smallest units of an image. They are tiny dots that collectively form an image on a screen. Their arrangement determines the image's resolution and quality.
In a digital image, each pixel carries data, be it in color or in intensity for grayscale. The more pixels there are, the clearer and more detailed an image appears. For instance, the resolution of 1,280 x 840 means the image contains 1,075,200 pixels in total, giving it its size and detail.

• Resolution: Determines total pixel count (e.g., 1,280 x 840).
• Quality: Higher pixel counts increase image clarity.
• Representation: Each pixel can be thought of as a data cell storing color or brightness in an image.

This understanding helps when calculating data that needs to be transmitted, as each pixel's information affects the total data size.

Bits Per Pixel

Bits per pixel is a measure of how many bits are used to represent each pixel's color or brightness in an image. More bits mean more detailed and richer color information when it comes to color images.
In a grayscale image, like in the given exercise, 8 bits per pixel are used. This allows each pixel to display 256 different shades of gray (from 0 to 255). The more bits per pixel, the more precision and finer the gradations of gray that can be displayed.

• 8 bits per pixel: Offers 256 shades of gray.
• Color depth: More bits provide richer image detail.
• Data size: Directly affects transmission size; more bits lead to larger data.

Understanding bits per pixel is crucial when dealing with image transmission, as it determines the size of the data that needs to be sent over a network.