Texas Instruments Inc reported higher-than-expected fourth-quarter revenue on Monday as demand improved faster than expected because customers did not have enough chips in stock, signaling the end of an inventory correction.

TI shares rose more than 3.4 percent after-hours. The company makes chips for a broad array of products ranging from consumer electronics to industrial equipment.

TI saw a broad resumption of demand across most of its products in December which continued into January, said Chief Financial Officer Kevin March.

We think our customers have reached a point where their (inventory) numbers were extremely lean, so they needed to order new chips quickly to meet demand, March told Reuters.

Before TI issued results, upbeat forecasts by smaller rivals Linear Technology and Xilinx last week had already made investors optimistic that a drawdown of chip inventories in the broader market may be ending.

Much of the chip industry has been in a funk since mid-2011, when manufacturers began to cut orders of new product and use up existing inventories due to jitters about a gloomy economy.

TI reported a fourth-quarter profit of $298 million, or 25 cents per share, compared with $942 million, or 78 cents per share, in the year-ago quarter.

Revenue fell to $3.42 billion from $3.53 billion, but above Wall Street's expectations of $3.25 billion, according to Thomson Reuters I/B/E/S.

TI had warned December 8 that chip demand was weak. It forecast fourth-quarter earnings per share in a range of 21 to 25 cents on revenue of $3.19 billion to $3.33 billion.

But March said that orders started to improve right away after the December update.

TI's forecast for first-quarter revenue in a range of $3.02 billion to $3.28 billion was much weaker than Wall Street expectations for $3.47 billion, according to Thomson Reuters I/B/E/S.

But March blamed TI's weak revenue guidance for the current quarter on a decline in its baseband chip segment, which it plans to shutter by the end of this year.

Revenue from the baseband business, whose main customer is Nokia , will fall to about $75 million this quarter from $279 million in the fourth quarter, March said.

Excluding the decline in baseband revenue, he said TI's first-quarter revenue would have fallen 2 percent from the fourth quarter, compared with the more typical sequential decline of 4 percent.

TI shares were up more than 3.4 percent at $34.34 in after-hours trading on Monday, compared with their Nasdaq closing price of $33.19.

(Reporting by Sinead Carew in New York; Editing by Steve Orlofsky and Matthew Lewis)