Consumer price index

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Consumer_price_index 

 

Consumer Price Index

  CPI

  Core CPI

 

  M2 % change from a year ago

  CPI

 

Inflation and federal funds rate

A consumer price index (CPI) is a price index, the price of a weighted average market basket of consumer goods and services purchased by households. Changes in measured CPI track changes in prices over time.

Overview

A CPI is a statistical estimate constructed using the prices of a sample of representative items whose prices are collected periodically. Sub-indices and sub-sub-indices can be computed for different categories and sub-categories of goods and services, being combined to produce the overall index with weights reflecting their shares in the total of the consumer expenditures covered by the index. It is one of several price indices calculated by most national statistical agencies. The annual percentage change in a CPI is used as a measure of inflation. A CPI can be used to index (i.e. adjust for the effect of inflation) the real value of wages, salaries, and pensions; to regulate prices; and to deflate monetary magnitudes to show changes in real values. In most countries, the CPI, along with the population census, is one of the most closely watched national economic statistics.

Inflation compared to federal funds rate

 

A graph of the US CPI from 1913 (in blue), and its percentage annual change (in red)

The index is usually computed monthly, or quarterly in some countries, as a weighted average of sub-indices for different components of consumer expenditure, such as food, housing, shoes, clothing, each of which is, in turn, a weighted average of sub-sub-indices. At the most detailed level, the elementary aggregate level, (for example, men's shirts sold in department stores in San Francisco), detailed weighting information is unavailable, so indices are computed using an unweighted arithmetic or geometric mean of the prices of the sampled product offers. (However, the growing use of barcode scanner data is gradually making weighting information available even at the most detailed level.) These indices compare prices each month with prices in the price-reference month. The weights used to combine them into the higher-level aggregates, and then into the overall index, relate to the estimated expenditures during a preceding whole year of the consumers covered by the index on the products within its scope in the area covered. Thus the index is a fixed-weight index, but rarely a true Laspeyres index, since the weight-reference period of a year and the price-reference period, usually a more recent single month, do not coincide.

Ideally, the weights would relate to the composition of expenditure during the time between the price-reference month and the current month. There is a large technical economics literature on index formulas which would approximate this and which can be shown to approximate what economic theorists call a true cost-of-living index. Such an index would show how consumer expenditure would have to move to compensate for price changes so as to allow consumers to maintain a constant standard of living. Approximations can only be computed retrospectively, whereas the index has to appear monthly and, preferably, quite soon. Nevertheless, in some countries, notably in the United States and Sweden, the philosophy of the index is that it is inspired by and approximates the notion of a true cost of living (constant utility) index, whereas in most of Europe it is regarded more pragmatically.

The coverage of the index may be limited. Consumers' expenditure abroad is usually excluded; visitors' expenditure within the country may be excluded in principle if not in practice; the rural population may or may not be included; certain groups such as the very rich or the very poor may be excluded. Saving and investment are always excluded, though the prices paid for financial services provided by financial intermediaries may be included along with insurance.

The index reference period, usually called the base year, often differs both from the weight-reference period and the price-reference period. This is just a matter of rescaling the whole time series to make the value for the index reference-period equal to 100. Annually revised weights are a desirable but expensive feature of an index, for the older the weights the greater is the divergence between the current expenditure pattern and that of the weight reference-period.

It is calculated and reported on a per region or country basis on a monthly and annual basis. International organizations like the Organisation for Economic Co-operation and Development (OECD) report statistical figures like the consumer price index for many of its member countries. In the US the CPI is usually reported by the Bureau of Labor Statistics.

An English economist by the name of Joseph Lowe first proposed the theory of price basket index in 1832. His fixed basket approach was relatively simple as Lowe computed the price of a list of goods in period 0 and compared the price of that same basket of goods in period 1. Since his proposed theories however were elementary, later economists built on his ideas to form our modern definition.

Calculating the CPI for a single item

${\displaystyle {\text{consumer price index}}={\frac {\text{market basket of desired year}}{\text{market basket of base year}}}\times {\text{100}}}$

or

${\displaystyle {\frac {{\text{CPI}}_{2}}{{\text{CPI}}_{1}}}={\frac {{\text{price}}_{2}}{{\text{price}}_{1}}}}$

Where 1 is usually the comparison year and CPI₁ is usually an index of 100.

Alternatively, the CPI can be performed as ${\displaystyle {\text{CPI}}={\frac {\text{updated cost}}{\text{base period cost}}}\times 100}$. The "updated cost" (i.e. the price of an item at a given year, e.g.: the price of bread in 2018) is divided by that of the initial year (the price of bread in 1970), then multiplied by one hundred.

Calculating the CPI for multiple items

Many but not all price indices are weighted averages using weights that sum to 1 or 100.

Example: The prices of 85,000 items from 22,000 stores, and 35,000 rental units are added together and averaged. They are weighted this way: housing 41.4%; food and beverages 17.4%; transport 17.0%; medical care 6.9%; apparel 6.0%; entertainment 4.4%; other 6.9%. Taxes (43%) are not included in CPI computation.

${\displaystyle \mathrm {CPI} ={\frac {\sum _{i=1}^{n}\mathrm {CPI} _{i}\times \mathrm {weight} _{i}}{\sum _{i=1}^{n}\mathrm {weight} _{i}}}}$

where the ${\displaystyle \mathrm {weight} _{i}}$ terms do not necessarily sum to 1 or 100.

Weighting

Weights and sub-indices

By convention, weights are fractions or ratios summing to one, as percentages summing to 100 or as per mille numbers summing to 1000.

On the European Union's Harmonized Index of Consumer Prices (HICP), for example, each country computes some 80 prescribed sub-indices, their weighted average constituting the national HICP. The weights for these sub-indices will consist of the sum of the weights of a number of component lower level indices. The classification is according to use, developed in a national accounting context. This is not necessarily the kind of classification that is most appropriate for a consumer price index. Grouping together of substitutes or of products whose prices tend to move in parallel might be more suitable.

For some of these lower-level indices detailed reweighing to make them be available, allowing computations where the individual price observations can all be weighted. This may be the case, for example, where all selling is in the hands of a single national organisation which makes its data available to the index compilers. For most lower level indices, however, the weight will consist of the sum of the weights of a number of elementary aggregate indices, each weight corresponding to its fraction of the total annual expenditure covered by the index. An 'elementary aggregate' is a lowest-level component of expenditure: this has a weight, but the weights of each of its sub-components are usually lacking. Thus, for example: Weighted averages of elementary aggregate indices (e.g. for men's shirts, raincoats, women's dresses, etc.) make up low-level indices (e.g. outer garments).

Weight averages of these, in turn, provide sub-indices at a higher, more aggregated level (e.g. clothing) and weighted averages of the latter provide yet more aggregated sub-indices (e.g. Clothing and Footwear).

Some of the elementary aggregate indices and some of the sub-indices can be defined simply in terms of the types of goods and/or services they cover. In the case of such products as newspapers in some countries and postal services, which have nationally uniform prices. But where price movements do differ or might differ between regions or between outlet types, separate regional and/or outlet-type elementary aggregates are ideally required for each detailed category of goods and services, each with its own weight. An example might be an elementary aggregate for sliced bread sold in supermarkets in the Northern region.

Most elementary aggregate indices are necessarily 'unweighted' averages for the sample of products within the sampled outlets. However, in cases where it is possible to select the sample of outlets from which prices are collected so as to reflect the shares of sales to consumers of the different outlet types covered, self-weighted elementary aggregate indices may be computed. Similarly, if the market shares of the different types of products represented by product types are known, even only approximately, the number of observed products to be priced for each of them can be made proportional to those shares.

Estimating weights

The outlet and regional dimensions noted above mean that the estimation of weights involves a lot more than just the breakdown of expenditure by types of goods and services, and the number of separately weighted indices composing the overall index depends upon two factors:

1. The degree of detail to which available data permit breakdown of total consumption expenditure in the weight reference-period by type of expenditure, region and outlet type.
2. Whether there is reason to believe that price movements vary between these most detailed categories.

How the weights are calculated, and in how much detail, depends upon the availability of information and upon the scope of the index. In the UK the retail price index (RPI) does not relate to the whole of consumption, for the reference population is all private households with the exception of pensioner households that derive at least three-quarters of their total income from state pensions and benefits, and "high income households" whose total household income lies within the top four per cent of all households. The result is that it is difficult to use data sources relating to total consumption by all population groups.

For products whose price movements can differ between regions and between different types of outlet:

• The ideal, rarely realizable in practice, would consist of estimates of expenditure for each detailed consumption category, for each type of outlet, for each region.
• At the opposite extreme, with no regional data on expenditure totals but only on population (e.g. 24% in the Northern region) and only national estimates for the shares of different outlet types for broad categories of consumption (e.g. 70% of food sold in supermarkets) the weight for sliced bread sold in supermarkets in the Northern region has to be estimated as the share of sliced bread in total consumption × 0.24 × 0.7.

The situation in most countries comes somewhere between these two extremes. The point is to make the best use of whatever data are available.

The nature of the data used for weighing

No firm rules can be suggested on this issue for the simple reason that the available statistical sources differ between countries. However, all countries conduct periodical household-expenditure surveys and all produce breakdowns of consumption expenditure in their national accounts. The expenditure classifications used there may however be different. In particular:

• Household-expenditure surveys do not cover the expenditures of foreign visitors, though these may be within the scope of a consumer price index.
• National accounts include imputed rents for owner-occupied dwellings which may not be within the scope of a consumer price index.

Even with the necessary adjustments, the national-account estimates and household-expenditure surveys usually diverge.

The statistical sources required for regional and outlet-type breakdowns are usually weak. Only a large-sample Household Expenditure survey can provide a regional breakdown. Regional population data are sometimes used for this purpose, but need adjustment to allow for regional differences in living standards and consumption patterns. Statistics of retail sales and market research reports can provide information for estimating outlet-type breakdowns, but the classifications they use rarely correspond to COICOP categories.

The increasingly widespread use of bar codes, scanners in shops has meant that detailed cash register printed receipts are provided by shops for an increasing share of retail purchases. This development makes possible improved Household Expenditure surveys, as Statistics Iceland has demonstrated. Survey respondents keeping a diary of their purchases need to record only the total of purchases when itemized receipts were given to them and keep these receipts in a special pocket in the diary. These receipts provide not only a detailed breakdown of purchases but also the name of the outlet. Thus response burden is markedly reduced, accuracy is increased, product description is more specific and point of purchase data are obtained, facilitating the estimation of outlet-type weights.

There are only two general principles for the estimation of weights: use all the available information and accept that rough estimates are better than no estimates.

Reweighing

Ideally, in computing an index, the weights would represent current annual expenditure patterns. In practice, they necessarily reflect past using the most recent data available or, if they are not of high quality, some average of the data for more than one previous year. Some countries have used a three-year average in recognition of the fact that household survey estimates are of poor quality. In some cases, some of the data sources used may not be available annually, in which case some of the weights for lower-level aggregates within higher-level aggregates are based on older data than the higher level weights.

Infrequent reweighing saves costs for the national statistical office but delays the introduction into the index of new types of expenditure. For example, subscriptions for Internet service entered index compilation with a considerable time lag in some countries, and account could be taken of digital camera prices between re-weightings only by including some digital cameras in the same elementary aggregate as film cameras.

Owner-occupiers and the price index

The way in which owner-occupied dwellings should be dealt with in a consumer price index has been, and remains, a subject of heated controversy in many countries. Various approaches have been considered, each with their advantages and disadvantages.

The economists' approach

See also: Imputed rent

Leaving aside the quality of public services, the environment, crime and so forth, and regarding the standard of living as a function of the level and composition of individuals' consumption, this standard depends upon the amount and range of goods and services they consume. These include the service provided by rented accommodation, which can readily be priced, and the similar services yielded by a flat or house owned by the consumer who occupies it. Its cost to a consumer is, according to the economic way of thinking, an "opportunity cost", namely what he or she sacrifices by living in it. This cost, according to many economists, is what should form a component of a consumer price index.

Opportunity cost can be looked at in two ways, since there are two alternatives to continuing to live in an owner-occupied dwelling. One – supposing that it is one year's cost that is to be considered – is to sell it, earn interest on the owner's capital thus released, and buy it back a year later, making an allowance for its physical depreciation. This can be called the "alternative cost" approach. The other, the "rental equivalent" approach, is to let it to someone else for the year, in which case the cost is the rent that could be obtained for it.

There are practical problems in implementing either of these economists' approaches. Thus, with the alternative cost approach, if house prices are rising fast the cost can be negative and then become sharply positive once house prices start to fall, so such an index would be very volatile. On the other hand, with the rental equivalent approach, there may be difficulty in estimating the movement of rental values of types of property that are not actually rented. If one or other of these measures of the consumption of the services of owner-occupied dwellings is included in consumption, then it must be included in income too, for income equals consumption plus saving. This means that if the movement of incomes is to be compared with the movement of the consumer price index, incomes must be expressed as money income plus this imaginary consumption value. That is logical, but it may not be what users of the index want.

Although the argument has been expressed in connection with owner-occupied dwellings, the logic applies equally to all durable consumer goods and services. Furniture, carpets and domestic appliances are not used up soon after purchase in the way that food is. Like dwellings, they yield a consumption service that can continue for years. Furthermore, since strict logic is to be adhered to, there are durable services as well that ought to be treated in the same way; the service consumers derive from appendectomies or crowned teeth continue for a long time. Since estimating values for these components of consumption has not been tackled, economic theorists are torn between their desire for intellectual consistency and their recognition that inclusion of the opportunity cost of the use of durables is impracticable.

Spending

Another approach is to concentrate on spending. Everyone agrees that repairs and maintenance expenditure of owner-occupied dwellings should be covered in a consumer price index, but the spending approach would include mortgage interest too. This turns out to be quite complicated, conceptually as well as in practice.

To explain what is involved, consider a consumer price index computed with reference to 2009 for just one sole consumer who bought her house in 2006, financing half of this sum by raising a mortgage. The problem is to compare how much interest such a consumer would now be paying with the interest that was paid in 2009. Since the aim is to compare like with like, that requires an estimate of how much interest would be paid now in the year 2010 on a similar house bought and 50% mortgage-financed three years ago, in 2007. It does not require an estimate of how much that identical person is paying now on the actual house she bought in 2006, even though that is what personally concerns her now.

A consumer price index compares how much it would cost now to do exactly what consumers did in the reference-period with what it cost then. Application of the principle thus requires that the index for our one house owner should reflect the movement of the prices of houses like hers from 2006 to 2007 and the change in interest rates. If she took out a fixed-interest rate mortgage it is the change in interest rates from 2006 to 2007 that counts; if she took out a variable interest mortgage it is the change from 2009 to 2010 that counts. Thus her current index with 1999 as reference-period will stand at more than 100 if house prices or, in the case of a fixed-interest mortgage, interest rates rose between 2006 and 2007.

The application of this principle in the owner-occupied dwellings component of a consumer price index is known as the "debt profile" method. It means that the current movement of the index will reflect past changes in dwelling prices and interest rates. Some people regard this as odd. Quite a few countries use the debt profile method, but in doing so most of them behave inconsistently. Consistency would require that the index should also cover the interest on consumer credit instead of the whole price paid for the products bought on credit if it covers mortgage interest payments. Products bought on credit would then be treated in the same way as owner-occupied dwellings.

Variants of the debt profile method are employed or have been proposed. One example is to include down payments as well as interest. Another is to correct nominal mortgage rates for changes in dwelling prices or for changes in the rest of the consumer price index to obtain a "real" rate of interest. Also, other methods may be used alongside the debt profile method. Thus several countries include a purely notional cost of depreciation as an additional index component, applying an arbitrarily estimated, or rather guessed, depreciation rate to the value of the stock of owner-occupied dwellings. Finally, one country includes both mortgage interest and purchase prices in its index.

Transaction prices

The third approach simply treats the acquisition of owner-occupied dwellings in the same way as acquisitions of other durable products are treated. This means:

• Taking account of the transaction prices agreed;
• Ignoring whether payments are delayed or are partly financed by borrowing;
• Leaving out second-hand transactions. Second-hand purchases correspond to sales by other consumers. Thus only new dwellings would be included.

Furthermore, expenditure on enlarging or reconstructing an owner-occupied dwelling would be covered, in addition to regular maintenance and repair. Two arguments of an almost theological character are advanced in connection with this transactions approach.

One argument is that purchases of new dwellings are treated as 'investment' in the system of national accounts, so should not enter a consumption price index. It is said that this is more than just a matter of terminological uniformity. For example, it may be thought to help understanding and facilitate economic analysis if what is included under the heading of 'consumption' is the same in the consumer price index and in the national income and expenditure accounts. Since these accounts include the equivalent rental value of owner-occupied dwellings, the equivalent rental approach would have to be applied in the consumer price index too. But the national accounts do not apply it to other durables, so the argument demands consistency in one respect but accepts its rejection in another.

The other argument is that the prices of new dwellings should exclude that part reflecting the value of the land, since this is an irreproducible and permanent asset that cannot be said to be consumed. This would presumably mean deducting site value from the price of a dwelling, site value presumably being defined as the price the site would fetch at auction if the dwelling were not on it. How this is to be understood in the case of multiple dwellings remains unclear.

Confusion

The merits of the different approaches are multidimensional, including feasibility, views on the way the index should and would move in particular circumstances, and theoretical properties of the index.

Statisticians in a country lacking a good dwelling price index (which is required for all except the rental equivalent method) will go along with a proposal to use such an index only if they can obtain the necessary additional resources that will enable them to compile one. Even obtaining mortgage interest rate data can be a major task in a country with a multitude of mortgage lenders and many types of mortgage. Dislike of the effect upon the behavior of the consumer price index arising from the adoption of some methods can be a powerful, if sometimes unprincipled, argument.

Dwelling prices are volatile and so, therefore, would be an index incorporating the current value of a dwelling price sub-index which, in some countries, would have a large weight under the third approach. Furthermore, the weight for owner-occupied dwellings could be altered considerably when reweighting was undertaken. (It could even become negative under the alternative cost approach if weights were estimated for a year during which house prices had been rising steeply).

Then, there is the point that a rise in interest rates designed to halt inflation could paradoxically make inflation appear higher if current interest rates showed up in the index. Economists' principles are not acceptable to all; nor is insistence upon consistency between the treatment of owner-occupied dwellings and other durables.

Consumer price indices in the United States

Main article: United States Consumer Price Index

In the United States several different consumer price indices are routinely computed by the Bureau of Labor Statistics (BLS). These include the CPI-U (for all urban consumers), CPI-W (for Urban Wage Earners and Clerical Workers), CPI-E (for the elderly), and C-CPI-U (chained CPI for all urban consumers). These are all built over two stages. First, the BLS collects data to estimate 8,018 separate item–area indices reflecting the prices of 211 categories of consumption items in 38 geographical areas. In the second stage, weighted averages are computed of these 8,018 item–area indices. The different indices differ only in the weights applied to the different 8,018 item–area indices. The weights for CPI-U and CPI-W are held constant for 24 months, changing in January of even-numbered years. The weights for C-CPI-U are updated each month to reflecting changes in consumption patterns in the last month. Thus, if people on average eat more chicken and less beef or more apples and fewer oranges than the previous month, that change would be reflected in next month's C-CPI-U. However, it would not be reflected in CPI-U and CPI-W until January of the next even-numbered year.

This allows the BLS to compute consumer price indices for each of the designated 38 geographical areas and for aggregates like the Midwest.

In January of each year, Social Security recipients receive a cost-of-living adjustment (COLA) "to ensure that the purchasing power of Social Security and Supplemental Security Income (SSI) benefits is not eroded by inflation. It is based on the percentage increase in the Consumer Price Index for Urban Wage Earners and Clerical Workers (CPI-W)". The use of CPI-W conflicts with this purpose, because the elderly consume substantially more health care goods and services than younger people. In recent years, inflation in health care has substantially exceeded inflation in the rest of the economy. Since the weight on health care in CPI-W is much less than the consumption patterns of the elderly, this COLA does not adequately compensate them for the real increases in the costs of the items they buy.

The BLS does track a consumer price index for the elderly (CPI-E). It is not used, in part because the social security trust fund is forecasted to run out of money in roughly 40 years, and using the CPI-E instead of CPI-W would shorten that by roughly five years.

The most recent December 2021 CPI reading hit 7%, the highest level in over 40 years. In response Jerome Powell, chair of the Federal Reserve has begun Quantitative tightening with rate hikes expected to begin in March 2022.

History

The CPI for various years are listed below with 1982 as the base year: A CPI of 150 means that there was 50% increase in prices, or 50% inflation, since 1982.

Year	CPI
1920	20.0
1930	16.7
1940	14.0
1950	24.1
1960	29.6
1970	38.8
1980	82.4
1982	100
1990	130.7
2000	172.2
2010	219.2
2018	251.1

Chained CPI

Main article: United States Chained Consumer Price Index

Former White House Chief of Staff Erskine Bowles and former U.S. Senator Alan K. Simpson suggested a transition to using a "chained CPI" in 2010, when they headed the White House's deficit-reduction commission. They stated that it was a more accurate measure of inflation than the current system and switching from the current system could save the government more than $290 billion over the decade following their report. "The chained CPI is usually 0.25 to 0.30 percentage points lower each year, on average, than the standard CPI measurements".

However, the National Active and Retired Federal Employees Associations said that the chained CPI does not account for seniors citizens' health care costs. Robert Reich, former United States Secretary of Labor under President Clinton, noted that typical seniors spend between 20 and 40 percent of their income on health care, far more than most Americans. "Besides, Social Security isn't in serious trouble. The Social Security trust fund is flush for at least two decades. If we want to ensure it's there beyond that, there's an easy fix – just lift the ceiling on income subject to Social Security taxes, which is now $113,700."

Replacing the current cost-of-living adjustment calculation with the chained CPI was considered, but not adopted, as part of a deficit-reduction proposal to avert the sequestration cuts, or fiscal cliff, in January 2013, but President Obama included it in his April 2013 budget proposal.

Personal consumption expenditures price index

Main article: Personal consumption expenditures price index

 

  CPI

  Core CPI

  PCE

  Core PCE

 

CPI vs PCE

Because of some shortcomings of the CPI, notably that it uses static expenditure weighting and it does not account for the substitution effect, the PCEPI is an alternative price index used by the Federal Reserve, among others, to measure inflation. From January 1959 through July 2018, inflation measured by the PCEPI has averaged 3.3%, while it has averaged 3.8% using CPI.

Pulse-Doppler radar

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Pulse-Doppler_radar 

 

Airborne pulse-Doppler radar antenna

A pulse-Doppler radar is a radar system that determines the range to a target using pulse-timing techniques, and uses the Doppler effect of the returned signal to determine the target object's velocity. It combines the features of pulse radars and continuous-wave radars, which were formerly separate due to the complexity of the electronics.

The first operational Pulse Doppler radar was in the CIM-10 Bomarc, an American long range supersonic missile powered by ramjet engines, and which was armed with a W40 nuclear weapon to destroy entire formations of attacking enemy aircraft. Pulse-Doppler systems were first widely used on fighter aircraft starting in the 1960s. Earlier radars had used pulse-timing in order to determine range and the angle of the antenna (or similar means) to determine the bearing. However, this only worked when the radar antenna was not pointed down; in that case the reflection off the ground overwhelmed any returns from other objects. As the ground moves at the same speed but opposite direction of the aircraft, Doppler techniques allow the ground return to be filtered out, revealing aircraft and vehicles. This gives pulse-Doppler radars "look-down/shoot-down" capability. A secondary advantage in military radar is to reduce the transmitted power while achieving acceptable performance for improved safety of stealthy radar.

Pulse-Doppler techniques also find widespread use in meteorological radars, allowing the radar to determine wind speed from the velocity of any precipitation in the air. Pulse-Doppler radar is also the basis of synthetic aperture radar used in radar astronomy, remote sensing and mapping. In air traffic control, they are used for discriminating aircraft from clutter. Besides the above conventional surveillance applications, pulse-Doppler radar has been successfully applied in healthcare, such as fall risk assessment and fall detection, for nursing or clinical purposes.

History

The earliest radar systems failed to operate as expected. The reason was traced to Doppler effects that degrade performance of systems not designed to account for moving objects. Fast-moving objects cause a phase-shift on the transmit pulse that can produce signal cancellation. Doppler has maximum detrimental effect on moving target indicator systems, which must use reverse phase shift for Doppler compensation in the detector.

Doppler weather effects (precipitation) were also found to degrade conventional radar and moving target indicator radar, which can mask aircraft reflections. This phenomenon was adapted for use with weather radar in the 1950s after declassification of some World War II systems.

Pulse-Doppler radar was developed during World War II to overcome limitations by increasing pulse repetition frequency. This required the development of the klystron, the traveling wave tube, and solid state devices. Early pulse-dopplers were incompatible with other high power microwave amplification devices that are not coherent, but more sophisticated techniques were developed that record the phase of each transmitted pulse for comparison to returned echoes.

Early examples of military systems includes the AN/SPG-51B developed during the 1950s specifically for the purpose of operating in hurricane conditions with no performance degradation.

The Hughes AN/ASG-18 Fire Control System was a prototype airborne radar/combination system for the planned North American XF-108 Rapier interceptor aircraft for the United States Air Force, and later for the Lockheed YF-12. The US's first pulse-Doppler radar, the system had look-down/shoot-down capability and could track one target at a time.

Weather, chaff, terrain, flying techniques, and stealth are common tactics used to hide aircraft from radar. Pulse-Doppler radar eliminates these weaknesses.

It became possible to use pulse-Doppler radar on aircraft after digital computers were incorporated in the design. Pulse-Doppler provided look-down/shoot-down capability to support air-to-air missile systems in most modern military aircraft by the mid 1970s.

Principle

Principle of pulse-Doppler radar

Range measurement

Principle of pulsed radar

Pulse-Doppler systems measure the range to objects by measuring the elapsed time between sending a pulse of radio energy and receiving a reflection of the object. Radio waves travel at the speed of light, so the distance to the object is the elapsed time multiplied by the speed of light, divided by two - there and back.

Velocity measurement

Change of wavelength caused by motion of the source

Pulse-Doppler radar is based on the Doppler effect, where movement in range produces frequency shift on the signal reflected from the target.

${\displaystyle {\text{Doppler frequency}}={\frac {2\times {\text{transmit frequency}}\times {\text{radial velocity}}}{C}}.}$

Radial velocity is essential for pulse-Doppler radar operation. As the reflector moves between each transmit pulse, the returned signal has a phase difference, or phase shift, from pulse to pulse. This causes the reflector to produce Doppler modulation on the reflected signal.

Pulse-Doppler radars exploit this phenomenon to improve performance.

The amplitude of the successively returning pulse from the same scanned volume is

${\displaystyle I=I_{0}\sin \left({\frac {4\pi (x_{0}+v\Delta t)}{\lambda }}\right)=I_{0}\sin(\Theta _{0}+\Delta \Theta ),}$

where

${\displaystyle x_{0}}$ is the distance radar to target,

${\displaystyle \lambda }$ is the radar wavelength,

${\displaystyle \Delta t}$ is the time between two pulses.

So

${\displaystyle \Delta \Theta ={\frac {4\pi v\Delta t}{\lambda }}.}$

This allows the radar to separate the reflections from multiple objects located in the same volume of space by separating the objects using a spread spectrum to segregate different signals:

${\displaystyle v={\text{target speed}}={\frac {\lambda \Delta \Theta }{4\pi \Delta t}},}$

where ${\displaystyle \Delta \Theta }$ is the phase shift induced by range motion.

Benefits

Rejection speed is selectable on pulse-Doppler aircraft-detection systems so nothing below that speed will be detected. A one degree antenna beam illuminates millions of square feet of terrain at 10 miles (16 km) range, and this produces thousands of detections at or below the horizon if Doppler is not used.

Pulse-Doppler radar uses the following signal processing criteria to exclude unwanted signals from slow-moving objects. This is also known as clutter rejection. Rejection velocity is usually set just above the prevailing wind speed (10 to 100 mile/hour or 15 to 150 km/hour). The velocity threshold is much lower for weather radar.

${\displaystyle \left\vert {\frac {{\text{Doppler frequency}}\times C}{2\times {\text{transmit frequency}}}}\right\vert >{\text{velocity threshold}}.}$

In airborne pulse-Doppler radar, the velocity threshold is offset by the speed of the aircraft relative to the ground.

${\displaystyle \left\vert {\frac {{\text{Doppler frequency}}\times C}{2\times {\text{transmit frequency}}}}-{\text{ground speed}}\times \cos \Theta \right\vert >{\text{velocity threshold}},}$

where ${\displaystyle \Theta }$ is the angle offset between the antenna position and the aircraft flight trajectory.

Surface reflections appear in almost all radar. Ground clutter generally appears in a circular region within a radius of about 25 miles (40 km) near ground-based radar. This distance extends much further in airborne and space radar. Clutter results from radio energy being reflected from the earth surface, buildings, and vegetation. Clutter includes weather in radar intended to detect and report aircraft and spacecraft.

Clutter creates a vulnerability region in pulse-amplitude time-domain radar. Non-Doppler radar systems cannot be pointed directly at the ground due to excessive false alarms, which overwhelm computers and operators. Sensitivity must be reduced near clutter to avoid overload. This vulnerability begins in the low-elevation region several beam widths above the horizon, and extends downward. This also exists throughout the volume of moving air associated with weather phenomenon.

Pulse-Doppler radar corrects this as follows.

• Allows the radar antenna to be pointed directly at the ground without overwhelming the computer and without reducing sensitivity.
• Fills in the vulnerability region associated with pulse-amplitude time-domain radar for small object detection near terrain and weather.
• Increases detection range by 300% or more in comparison to moving target indication (MTI) by improving sub-clutter visibility.

Clutter rejection capability of about 60 dB is needed for look-down/shoot-down capability, and pulse-Doppler is the only strategy that can satisfy this requirement. This eliminates vulnerabilities associated with the low-elevation and below-horizon environment.

Pulse compression, and moving target indicator (MTI) provide up to 25 dB sub-clutter visibility. MTI antenna beam is aimed above horizon to avoid excessive false alarm rate, which renders systems vulnerable. Aircraft and some missiles exploit this weakness using a technique called flying below the radar to avoid detection (Nap-of-the-earth). This flying technique is ineffective against pulse-Doppler radar.

Pulse-Doppler provides an advantage when attempting to detect missiles and low observability aircraft flying near terrain, sea surface, and weather.

Audible Doppler and target size support passive vehicle type classification when identification friend or foe is not available from a transponder signal. Medium pulse repetition frequency (PRF) reflected microwave signals fall between 1,500 and 15,000 cycle per second, which is audible. This means a helicopter sounds like a helicopter, a jet sounds like a jet, and propeller aircraft sound like propellers. Aircraft with no moving parts produce a tone. The actual size of the target can be calculated using the audible signal.

Detriments

Maximum range from reflectivity (red) and unambiguous Doppler velocity range (blue) with a fix pulse repetition rate.

Ambiguity processing is required when target range is above the red line in the graphic, which increases scan time.

Scan time is a critical factor for some systems because vehicles moving at or above the speed of sound can travel one mile (1.6 km) every few seconds, like the Exocet, Harpoon, Kitchen, and Air-to-air missile. The maximum time to scan the entire volume of the sky must be on the order of a dozen seconds or less for systems operating in that environment.

Pulse-Doppler radar by itself can be too slow to cover the entire volume of space above the horizon unless fan beam is used. This approach is used with the AN/SPS 49(V)5 Very Long Range Air Surveillance Radar, which sacrifices elevation measurement to gain speed.

Pulse-Doppler antenna motion must be slow enough so that all the return signals from at least 3 different PRFs can be processed out to the maximum anticipated detection range. This is known as dwell time. Antenna motion for pulse-Doppler must be as slow as radar using MTI.

Search radar that include pulse-Doppler are usually dual mode because best overall performance is achieved when pulse-Doppler is used for areas with high false alarm rates (horizon or below and weather), while conventional radar will scan faster in free-space where false alarm rate is low (above horizon with clear skies).

The antenna type is an important consideration for multi-mode radar because undesirable phase shift introduced by the radar antenna can degrade performance measurements for sub-clutter visibility.

Signal processing

The signal processing enhancement of pulse-Doppler allows small high-speed objects to be detected in close proximity to large slow moving reflectors. To achieve this, the transmitter must be coherent and should produce low phase noise during the detection interval, and the receiver must have large instantaneous dynamic range.

• Pulse-Doppler signal processing detailed explanation

Pulse-Doppler signal processing also includes ambiguity resolution to identify true range and velocity.

• Ambiguity resolution detailed explanation

The received signals from multiple PRF are compared to determine true range using the range ambiguity resolution process.

• Range ambiguity resolution detailed explanation

The received signals are also compared using the frequency ambiguity resolution process.

• Frequency ambiguity resolution detailed explanation

Range resolution

The range resolution is the minimal range separation between two objects traveling at the same speed before the radar can detect two discrete reflections:

${\displaystyle {\text{range resolution}}={\frac {C}{{\text{PRF}}\times ({\text{number of samples between transmit pulses}})}}.}$

In addition to this sampling limit, the duration of the transmitted pulse could mean that returns from two targets will be received simultaneously from different parts of the pulse.

Velocity resolution

The velocity resolution is the minimal radial velocity difference between two objects traveling at the same range before the radar can detect two discrete reflections:

${\displaystyle {\text{velocity resolution}}={\frac {C\times {\text{PRF}}}{2\times {\text{transmit frequency}}\times {\text{filter size in transmit pulses}}}}.}$

Special consideration

Pulse-Doppler radar has special requirements that must be satisfied to achieve acceptable performance.

Pulse repetition frequency

Main article: Pulse repetition frequency

Pulse-Doppler typically uses medium pulse repetition frequency (PRF) from about 3 kHz to 30 kHz. The range between transmit pulses is 5 km to 50 km.

Range and velocity cannot be measured directly using medium PRF, and ambiguity resolution is required to identify true range and speed. Doppler signals are generally above 1 kHz, which is audible, so audio signals from medium-PRF systems can be used for passive target classification.

Angular measurement

Radar systems require angular measurement. Transponders are not normally associated with pulse-Doppler radar, so sidelobe suppression is required for practical operation.

Tracking radar systems use angle error to improve accuracy by producing measurements perpendicular to the radar antenna beam. Angular measurements are averaged over a span of time and combined with radial movement to develop information suitable to predict target position for a short time into the future.

The two angle error techniques used with tracking radar are monopulse and conical scan.

• Monopulse radar
• Conical scanning

Coherency

Pulse-Doppler radar requires a coherent oscillator with very little noise. Phase noise reduces sub-clutter visibility performance by producing apparent motion on stationary objects.

Cavity magnetron and crossed-field amplifier are not appropriate because noise introduced by these devices interfere with detection performance. The only amplification devices suitable for pulse-Doppler are klystron, traveling wave tube, and solid state devices.

Scalloping

Main article: Radar scalloping

Pulse-Doppler signal processing introduces a phenomenon called scalloping. The name is associated with a series of holes that are scooped-out of the detection performance.

Scalloping for pulse-Doppler radar involves blind velocities created by the clutter rejection filter. Every volume of space must be scanned using 3 or more different PRF. A two PRF detection scheme will have detection gaps with a pattern of discrete ranges, each of which has a blind velocity.

Windowing

Ringing artifacts pose a problem with search, detection, and ambiguity resolution in pulse-Doppler radar.

Ringing is reduced in two ways.

First, the shape of the transmit pulse is adjusted to smooth the leading edge and trailing edge so that RF power is increased and decreased without an abrupt change. This creates a transmit pulse with smooth ends instead of a square wave, which reduces ringing phenomenon that is otherwise associated with target reflection.

Second, the shape of the receive pulse is adjusted using a window function that minimizes ringing that occurs any time pulses are applied to a filter. In a digital system, this adjusts the phase and/or amplitude of each sample before it is applied to the fast Fourier transform. The Dolph-Chebyshev window is the most effective because it produces a flat processing floor with no ringing that would otherwise cause false alarms.

Antenna

Pulse-Doppler radar is generally limited to mechanically aimed antennas and active phase array.

Mechanical RF components, such as wave-guide, can produce Doppler modulation due to phase shift induced by vibration. This introduces a requirement to perform full spectrum operational tests using shake tables that can produce high power mechanical vibration across all anticipated audio frequencies.

Doppler is incompatible with most electronically steered phase-array antenna. This is because the phase-shifter elements in the antenna are non-reciprocal and the phase shift must be adjusted before and after each transmit pulse. Spurious phase shift is produced by the sudden impulse of the phase shift, and settling during the receive period between transmit pulses places Doppler modulation onto stationary clutter. That receive modulation corrupts the measure of performance for sub-clutter visibility. Phase shifter settling time on the order of 50ns is required. Start of receiver sampling needs to be postponed at least 1 phase-shifter settling time-constant (or more) for each 20 dB of sub-clutter visibility.

Most antenna phase shifters operating at PRF above 1 kHz introduce spurious phase shift unless special provisions are made, such as reducing phase shifter settling time to a few dozen nanoseconds.

The following gives the maximum permissible settling time for antenna phase shift modules.

${\displaystyle T={\frac {1}{e^{\frac {\text{SCV}}{20}}\times S\times {\text{PRF}}}},}$

where

T = phase shifter settling time,

SCV = sub-clutter visibility in dB,

S = number of range samples between each transmit pulse,

PRF = maximal design pulse repetition frequency.

The antenna type and scan performance is a practical consideration for multi-mode radar systems.

Diffraction

Choppy surfaces, like waves and trees, form a diffraction grating suitable for bending microwave signals. Pulse-Doppler can be so sensitive that diffraction from mountains, buildings or wave tops can be used to detect fast moving objects otherwise blocked by solid obstruction along the line of sight. This is a very lossy phenomenon that only becomes possible when radar has significant excess sub-clutter visibility.

Refraction and ducting use transmit frequency at L-band or lower to extend the horizon, which is very different from diffraction. Refraction for over-the-horizon radar uses variable density in the air column above the surface of the earth to bend RF signals. An inversion layer can produce a transient troposphere duct that traps RF signals in a thin layer of air like a wave-guide.

Subclutter visibility

Subclutter visibility involves the maximum ratio of clutter power to target power, which is proportional to dynamic range. This determines performance in heavy weather and near the earth surface.

${\displaystyle {\text{dynamic range}}=\min {\begin{cases}{\tfrac {\text{carrier power}}{\text{noise power}}}&{\text{transmit noise, where bandwidth is }}{\tfrac {\text{PRF}}{\text{filter size}}}\\2^{{\text{sample bits}}+{\text{filter size}}}&{\text{receiver dynamic range}}\end{cases}}.}$

Subclutter visibility is the ratio of the smallest signal that can be detected in the presence of a larger signal.

${\displaystyle {\text{subclutter visibility}}={\frac {\text{dynamic range}}{\text{CFAR detection threshold}}}.}$

A small fast-moving target reflection can be detected in the presence of larger slow-moving clutter reflections when the following is true:

${\displaystyle {\text{target power}}>{\frac {\text{clutter power}}{\text{subclutter visibility}}}.}$

Performance

The pulse-Doppler radar equation can be used to understand trade-offs between different design constraints, like power consumption, detection range, and microwave safety hazards. This is a very simple form of modeling that allows performance to be evaluated in a sterile environment.

The theoretical range performance is as follows.

${\displaystyle R=\left({\frac {P_{t}G_{t}A_{r}\sigma FD}{16\pi ^{2}K_{b}TBN}}\right)^{\frac {1}{4}},}$

where

R = distance to the target,

P_t = transmitter power,

G_t = gain of the transmitting antenna,

A_r = effective aperture (area) of the receiving antenna,

σ = radar cross section, or scattering coefficient, of the target,

F = antenna pattern propagation factor,

D = Doppler filter size (transmit pulses in each Fast Fourier transform),

K_b = Boltzmann's constant,

T = absolute temperature,

B = receiver bandwidth (band-pass filter),

N = noise figure.

This equation is derived by combining the radar equation with the noise equation and accounting for in-band noise distribution across multiple detection filters. The value D is added to the standard radar range equation to account for both pulse-Doppler signal processing and transmitter FM noise reduction.

Detection range is increased proportional to the fourth root of the number of filters for a given power consumption. Alternatively, power consumption is reduced by the number of filers for a given detection range.

Pulse-Doppler signal processing integrates all of the energy from all of the individual reflected pulses that enter the filter. This means a pulse-Doppler signal processing system with 1024 elements provides 30.103 dB of improvement due to the type of signal processing that must be used with pulse-Doppler radar. The energy of all of the individual pulses from the object are added together by the filtering process.

Signal processing for a 1024-point filter improves performance by 30.103 dB, assuming compatible transmitter and antenna. This corresponds to 562% increase in maximal distance.

These improvements are the reason pulse-Doppler is essential for military and astronomy.

Aircraft tracking uses

Pulse-Doppler radar for aircraft detection has two modes.

• Scan
• Track

Scan mode involves frequency filtering, amplitude thresholding, and ambiguity resolution. Once a reflection has been detected and resolved, the pulse-Doppler radar automatically transitions to tracking mode for the volume of space surrounding the track.

Track mode works like a phase-locked loop, where Doppler velocity is compared with the range movement on successive scans. Lock indicates the difference between the two measurements is below a threshold, which can only occur with an object that satisfies Newtonian mechanics. Other types of electronic signals cannot produce a lock. Lock exists in no other type of radar.

The lock criterion needs to be satisfied during normal operation.

• Pulse-Doppler signal processing § Lock

Lock eliminates the need for human intervention with the exception of helicopters and electronic jamming.

Weather phenomenon obey adiabatic process associated with air mass and not Newtonian mechanics, so the lock criterion is not normally used for weather radar.

Pulse-Doppler signal processing selectively excludes low-velocity reflections so that no detections occurs below a threshold velocity. This eliminates terrain, weather, biologicals, and mechanical jamming with the exception of decoy aircraft.

The target Doppler signal from the detection is converted from frequency domain back into time domain sound for the operator in track mode on some radar systems. The operator uses this sound for passive target classification, such as recognizing helicopters and electronic jamming.

Helicopters

Special consideration is required for aircraft with large moving parts because pulse-Doppler radar operates like a phase-locked loop. Blade tips moving near the speed of sound produce the only signal that can be detected when a helicopter is moving slow near terrain and weather.

Helicopters appears like a rapidly pulsing noise emitter except in a clear environment free from clutter. An audible signal is produced for passive identification of the type of airborne object. Microwave Doppler frequency shift produced by reflector motion falls into the audible sound range for human beings (20 – 20,000 Hz), which is used for target classification in addition to the kinds of conventional radar display used for that purpose, like A-scope, B-scope, C-scope, and RHI indicator. The human ear may be able to tell the difference better than electronic equipment.

A special mode is required because the Doppler velocity feedback information must be unlinked from radial movement so that the system can transition from scan to track with no lock.

Similar techniques are required to develop track information for jamming signals and interference that cannot satisfy the lock criterion.

Multi-mode

Pulse-Doppler radar must be multi-mode to handle aircraft turning and crossing trajectory.

Once in track mode, pulse-Doppler radar must include a way to modify Doppler filtering for the volume of space surrounding a track when radial velocity falls below the minimum detection velocity. Doppler filter adjustment must be linked with a radar track function to automatically adjust Doppler rejection speed within the volume of space surrounding the track.

Tracking will cease without this feature because the target signal will otherwise be rejected by the Doppler filter when radial velocity approaches zero because there is no change in frequency.

Multi-mode operation may also include continuous wave illumination for semi-active radar homing.