High Definition Television Bandwidth and Compression Math

Quote of the Day

What is the difference between an introverted and extroverted mathematician? An introverted mathematician stares at his shoes when you are talking to him. An extroverted mathematician stares at your shoes when you are talking to him.

— I heard this joke on the radio.


Introduction

Figure 1: HDTV Example. (Source)

Figure 1: HDTV Example. (Source)

I frequently hear people make comments about the data bandwidth required for HDTV on home data networks and the levels of compression used for HDTV broadcasts. I usually hear two numbers cited:

These numbers are important for engineers because data bandwidth drives the design of networks, and compression ratios drive image quality. I thought it might be useful to review the bandwidth requirements of HDTV and the level of compression involved. I looked around on the web for this data and I could not find the level of detail that I wanted -- I will put it here.

As part of my job, I actually encountered one person who regularly watches 14 simultaneous HDTV feeds. He is a stock trader who works out of his home. He was displaying stock data from multiple markets on multiple televisions. Think about it -- 14 · 20 Mbps = 280 Mbps of television being watched by one person.

Background

What is a Pixel?

A pixel is the smallest addressable unit on a video display. Figure 2 (Source) illustrates how each pixel is composed of three color-generating components, called sub-pixels. At the display, each pixel is commanded to generate a combination of different red (R), green (G), and blue (B) color intensities.

Figure 1: Pixel Configuration on an LED HDTV.

Figure 2: Pixel Configuration on an LED HDTV.

Figure 3 illustrates how the different color intensities can be used to make different colors.

Figure 2: Examples of Color Generation Using Pixels.

Figure 3: Examples of Color Generation Using Pixels.

Figure 4 shows how a simple color image is formed by varying the pixel color mixture.

Figure 3: Close-Up Image of Microsoft Flag showing Pixel Settings.

Figure 4: Close-Up Image of Microsoft Flag showing Pixel Settings.

What are Luma and Chroma?

At the display, we need RGB intensities in order to drive the sub-pixels. It turns out that RGB is not the most bandwidth efficient way to transfer pixel information. It has been recognized since the early days of television that a more bandwidth efficient approach is to restate RGB in in terms of luma and chroma, which are defined as follows:

  • luma

    Luma represents the brightness in an image -- the black and white portion of an image. It expressed as a linear combination of the red, green, and blue components. Symbolically, luma is expressed as Y. It turns out the the eye is very sensitive to differences in luma. Television acknowledges the sensitivity of eye to luma by giving it preferred fidelity over chroma. Mathematically, luma is expressed (per Rec. 709) as

    \text{Y }=\text{ }0.\text{2126 R }+\text{ }0.\text{7152 G }+\text{ }0.0\text{722 B}

    Luma also allows us to easily generate a signal for black and white-only televisions, which is important for supporting legacy television hardware.

  • chroma (sometimes called CbCr)

    Chroma represents the color portion of an image. It turns out that the eye does not have as fine a color resolution as it does to differences in light and darkness. HDTV, through a process referred to as chroma sub-sampling, reduces its bandwidth requirements by making pairs of pixels share chroma values.

    Chroma consists of two components.

    • Cb

      Sometimes called the "blue difference," it is a scaled and shifted version B − Y.

    • Cr

      Sometimes called the "red difference," it is a scaled and shifted version of R − Y.

In summary, the advantages of representing RGB in terms of luma and chroma are simple:

  • low-end, black and white only systems can easily display Y and ignore CbCr.
  • to reduce bandwidth requirements, we can broadcast a luma value for each pixel and we can share a CbCr value between pixels.

Display Characteristics

In North America, calling a televised image "HDTV" generally means that the image has the following characteristics:

  • an aspect ratio of 16 by 9.

    I must admit that I really do like HDTV's 16-to-9 aspect ratio over standard television's 4-to-3 aspect ratio. I am a big fan of old movies that were photographed in wide-screen formats like VistaVision and CinemaScope. HDTV makes watching these movies much more enjoyable. Unfortunately, more lines means more bandwidth.

  • interlaced scanning.

    North American standard definition television projects images at a 60 Hz rate. While the HDTV standard does support projecting a new image at 60 Hz, all the systems I know of reduce their bandwidth requirements by projecting the same image twice. This means that a new image is projected at a 30 Hz rate, which reduces the bandwidth required by half. This approach is referred to as 1080i/30.

  • the refresh rate is really 29.97 Hz

    In North America, we typically refer to 30 Hz and 60Hz refresh rates. These rates are only approximately correct. Historically, North American HDTV has used a frame rate of 29.97 Hz (=30 Hz/1.001, a number which would require an entire post to explain). In truth, our HDTV should be referred to as 1080p/29.97.

  • transmit 8 bits of luma and 8 bits of chroma (alternating between Cb and Cr) per pixel (Reference)

    Each pixel also requires 24 bits of color information: 8 bits of Cb , 8 bits of Cr, and 8 bits of Y. Luma and 1/2 the chroma information are passed with every pixel transferred. This means adjacent pixels must share a chroma value.

We can use this information to compute the total number of pixels using the method shown in Figure 5.

Figure 4: Calculation of the Total Number of HDTV Pixels.

Figure 5: Calculation of the Total Number of HDTV Pixels.

Analysis

HDTV Transmission Over a Wire Pair

Given the discussion above, we can calculate the HDTV bandwidth at the camera (i.e. wire-transfer) as shown in Figure 6.

Figure 5: Calculation Summary for HDTV Bandwidth Without Compression.

Figure 6: Calculation Summary for HDTV Bandwidth Without Compression.

So Figure 6 tells us that the raw bandwidth out of a camera is just about 1 Gbps.

HDTV Broadcasting

There is some basic information needed to understand how HDTV is broadcast. First, everything is packed into a 6 MHz bandwidth, which is the bandwidth of a standard North American television channel. Because some of that bandwidth is used for a pilot tone and for filter transition regions, only part of the 6 Mhz bandwidth can be used for data. The basic bandwidth allocation is illustrated in Figure 7.

Figure 6: Allocation of Image Bandwidth in a 6 MHz Channel.

Figure 7: Allocation of Image Bandwidth in a 6 MHz Channel.

Television standards state that the total bandwidth (i.e. information + pilot tone + transition regions) of the television channel is 11.5% greater than the information bandwidth. The 11.5% factor is usually called α. This number goes way back to the old NTSC standard.

Once we have all the pixel data, there is some overhead involved in packing the image data into the 8VSB transport protocol. This overhead is illustrated in Figure 8.

Figure 7: Data Frame Structure for HDTV Data.

Figure 8: Data Frame Structure for HDTV Data.

Figure 9 illustrates the actual calculation of the actual image bits available from a broadcast HDTV picture.

Figure 8: HDTV Broadcast Bit Rate Calculation.

Figure 9: HDTV Broadcast Bit Rate Calculation.

Compression Rates

Since we have the broadcast data rate (19.3 Mbps) and the raw data rate (994 Mbps), we can state that the actual compression ratio = 994 Mbps/19.3 Mpbs = 51.5 ≈ 50. So I think we have established where the 50-to-1 compression ratio comment comes from.

Conclusion

I was able to show the assumptions involved when people state that HDTV requires 19.3 Mbps of data bandwidth and it has a compression ratio of about 50. If you look carefully at the calculations, you can see that HDTV bandwidth will be increasing in the future as the color sampling resolution increases from 8-bits to 10-bits and the real refresh rate increases to 60 Hz.

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4 Responses to High Definition Television Bandwidth and Compression Math

  1. Fascinating! Your blog always does great job of unraveling the magic of the engineered world!

     
    • mathscinotes says:

      I see you are from PTC. Thanks for making a product like Mathcad. Engineers always have interesting problems and solutions -- the problem is getting it all written down. That is why I am such a big proponent of Mathcad. I have found it to be a great vehicle for the quick dissemination of both information and tools. Keep up the good work. I know how tough product development and support can be.

      mathscinotes

       
  2. Pingback: Television Resolution | Math Encounters Blog

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